Method of measuring a physical function using a composite function which includes the physical function and an arbitrary reference function

ABSTRACT

A method for measuring a physical function forms a symmetric composite function by combining the physical function with a reference function. The method obtains a Fourier transform of the symmetric composite function. The method calculates an inverse Fourier transform of the obtained Fourier transform, wherein the calculated inverse Fourier transform provides information regarding the physical function. The physical function can be a nonlinearity profile of a sample with at least one sample surface. The physical function can alternatively by a sample temporal waveform of a sample optical pulse.

CLAIM OF PRIORITY

This application is a continuation-in-part of U.S. patent applicationSer. No. 10/378,591, filed Mar. 3, 2003 now U.S. Pat. No. 6,856,393,which is a continuation-in-part of U.S. patent application Ser. No.10/357,275, filed Jan. 31, 2003, which claims the benefit of and U.S.Provisional Application No. 60/405,405, filed Aug. 21, 2002. U.S. patentapplication Ser. No. 10/378,591, U.S. patent application Ser. No.10/357,275, and U.S. Provisional Application No. 60/405,405, filed Aug.21, 2002, are incorporated by reference in their entireties herein.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates generally to the field of measuringattributes of a physical system and, more particularly, relates tomethods of measuring a non-symmetric physical function.

2. Description of the Related Art

Various optical devices are based on induced second-ordersusceptibilities in silica-based glass waveguides (e.g., electro-opticmodulators, switches, parametric amplifiers). For example, G. Bonfrateet al. describe optical parametric oscillators useful for the study ofquantum interference and quantum cryptography, and for metrologyapplications in Parametric Fluorescence in Periodically Poled SilicaFibers, Applied Physics Letters, Vol. 75, No. 16, 1999, pages 2356–2358,which is incorporated in its entirety by reference herein. Second-ordersusceptibility can be induced in a several-microns-thick region of fusedsilica (a material that is not normally centro-symmetric, and thereforenormally does not exhibit a second-order susceptibility) by poling atelevated temperatures. This phenomenon has been described by R. A. Myerset al. in Large Second-Order Nonlinearity in Poled Fused Silica, OpticsLetters, Vol. 16, No. 22, 1991, pages 1732–1734, which is incorporatedin its entirety by reference herein.

FIGS. 1A and 1B schematically illustrate the poling of a silica wafer 1.As schematically illustrated in FIG. 1A, poling typically comprisesusing an anode electrode 2 placed proximate to one surface 3 of thewafer 1 and a cathode electrode 4 placed proximate to the oppositesurface 5 of the wafer 1. A voltage is applied across the wafer 1 for aperiod of time, resulting in a second-order optical nonlinearityprofile. The profile has a thickness and is localized beneath thesurface 3 where the anode electrode was placed, as schematicallyillustrated in FIG. 1B. As used herein, the term “anodic surface” refersto the surface which is placed proximate to the anode electrode, and theterm “cathodic surface” refers to the surface which is placed proximateto the cathode electrode. Such a poling procedure is described in moredetail by Thomas G. Alley et al. in Space Charge Dynamics in ThermallyPoled Fused Silica, Journal of Non-Crystalline Solids, Vol. 242, 1998,pages 165–176, which is incorporated herein in its entirety.

The field of poled silica has suffered from the lack of a common methodto reliably measure the second-order optical nonlinearity profile ofpoled samples. This absence of a reliable procedure for measuringnonlinearity profiles may be the reason, at least in part, for widediscrepancies in the measured magnitudes and the positions of thenonlinearity profiles of various poled systems as reported in theliterature. The Maker fringe (MF) technique is the most common methodcurrently used to investigate the nonlinearity profile of poled silica.The MF technique comprises focusing a pulsed laser beam of intensity I₁(known as the fundamental signal) onto a sample at an incident angle θand measuring the intensity I₂ of the second harmonic (SH) signalgenerated within the nonlinear region as a function of the incidentangle θ. For a transverse magnetic (TM) polarized fundamental laserbeam, the conversion efficiency η_(TM)(θ) is given by:

$\begin{matrix}{{\eta_{TM}(\theta)} = {\frac{I_{2}}{I_{1}} = \left. {f\left( {\theta,n_{1},n_{2}} \right)} \middle| {\int_{\;}^{\;}\;{{\mathbb{d}_{33}(z)}{\mathbb{e}}^{j\;\Delta\;{k{(\theta)}}z}{\mathbb{d}z}}} \right|^{2}}} & (1)\end{matrix}$where

-   -   d₃₃(z) is the nonlinear coefficient (which is proportional to        the second-order susceptibility χ⁽²⁾);    -   z is the direction normal to the sample surface (i.e., parallel        to the poling field);    -   n₁ and n₂ are the refractive indices at the fundamental and SH        frequencies, respectively;    -   Δk=k₂−2k₁, where k₁ and k₂ are the fundamental and SH wave        numbers, respectively, and    -   ƒ(θn₁, n₂) is a well-defined function of the incident angle θ        (relative to the surface normal direction) and refractive        indices n₁ and n₂.

The function ƒ(θ, n₁, n₂) accounts for both the power loss due toreflection suffered by the fundamental and the SH beams, and theprojection of the input electric field along the appropriate direction.In general, ƒ(θ, n₁, n₂) depends on both the polarization of the inputfundamental wave and the geometry of the second harmonic generationconfiguration. The exact formula of ƒ(θ, n₁, n₂) is given by D. Pureur,et al. in Absolute Measurement of the Second-Order Nonlinearity Profilein Poled Silica, Optics Letters, Vol. 23, 1998, pages 588–590, which isincorporated in its entirety by reference herein. This phenomenon isalso described by P. D. Maker et al. in Effects of Dispersion andFocusing on the Production of Optical Harmonics, Physics Review Letters,Vol. 8, No. 1, 1962, pages 21–22, which is incorporated in its entiretyby reference herein.

The conversion efficiency η_(TM)(θ) is obtained experimentally byrotating the sample with respect to the incident laser beam andmeasuring the power of the SH signal as a function of the incident angleθ. Due to dispersion of the laser beam, Δk is finite and η_(TM)(θ)exhibits oscillations (called the Maker fringes) which pass throughseveral maxima and minima. The objective of this measurement is toretrieve the second-order nonlinearity profile d₃₃(z). The absolutevalue of the integral in Equation 1 is the amplitude of the Fouriertransform of d₃₃(z). In principle, if both the amplitude and the phaseof a Fourier transform are known, the argument of the Fourier transform(in this case d₃₃(z)) can be readily inferred by taking the inverseFourier transform of the Fourier transform. However, the measured Makerfringes provide only the magnitude of the Fourier transform, not itsphase. Consequently, for an arbitrary and unknown nonlinearity profile,the MF measurement alone is not sufficient to determine a uniquesolution for d₃₃(z). Even if the phase information were available, theshape of d₃₃(z) could be determined, but the location of this shapebeneath the surface of the sample (i.e., where the nonlinearity profilestarts beneath the surface) could not be determined.

Previous efforts to determine d₃₃(z) have involved fitting various trialprofiles to the measured MF data. Examples of such efforts are describedby M. Qiu et al. in Double Fitting of Maker Fringes to CharacterizeNear-Surface and Bulk Second-Order Nonlinearities in Poled Silica,Applied Physics Letters, Vol. 76, No. 23, 2000, pages 3346–3348; Y.Quiquempois et al. in Localisation of the Induced Second-OrderNon-Linearity Within Infrasil and Suprasil Thermally Poled Glasses,Optics Communications, Vol. 176, 2000, pages 479–487; and D. Faccio etal. in Dynamics of the Second-Order Nonlinearity in Thermally PoledSilica Glass, Applied Physics Letters, Vol. 79, No. 17, 2001, pages2687–2689. These references are incorporated in their entirety byreference herein.

However, the previous methods do not produce a unique solution ford₃₃(z). Two rather different trial profiles can provide almost equallygood fits to the measured MF data. This aspect of using fitting routinesto determine d₃₃(z) is described in more detail by Alice C. Liu et al.in Advances in the Measurement of the Poled Silica Nonlinear Profile,SPIE Conference on Doped Fiber Devices II, Boston, Mass., November 1998,pages 115–119, which is incorporated in its entirety by referenceherein.

SUMMARY OF THE INVENTION

According to one aspect of the present invention, a method measures aphysical function. The method comprises forming a symmetric compositefunction by combining the physical function with a reference function.The method further comprises obtaining a Fourier transform of thesymmetric composite function. The method further comprises calculatingan inverse Fourier transform of the obtained Fourier transform. Thecalculated inverse Fourier transform provides information regarding thephysical function.

In another aspect of the present invention, a method measures anonlinearity profile of a sample. In accordance with the method, asample having a sample nonlinearity profile is provided. The surface ofthe sample is placed in proximity to a surface of a supplemental sampleto form a composite sample having a composite nonlinearity profile. Themethod measures a Fourier transform magnitude of composite nonlinearityprofile, and calculates the sample nonlinearity profile using theFourier transform magnitude of the composite nonlinearity profile.

In still another aspect of the present invention, a method measures anonlinearity profile of a sample. In accordance with the method, asample is provided that has at least one sample surface and that has asample nonlinearity profile along a sample line through a predeterminedpoint on the sample surface. The sample line is oriented perpendicularlyto the sample surface. The method measures a Fourier transform magnitudeof the sample nonlinearity profile. The method provides a referencematerial having at least one reference surface and having a referencenonlinearity profile along a reference line through a predeterminedpoint on the reference surface. The reference line is orientedperpendicularly to the reference surface. The method obtains a Fouriertransform magnitude of the reference nonlinearity profile. The methodforms a first composite sample having a first composite nonlinearityprofile by placing the sample and the reference material proximate toone another in a first configuration with the sample line substantiallycollinear with the reference line. The method measures a Fouriertransform magnitude of the first composite nonlinearity profile. Themethod forms a second composite sample having a second compositenonlinearity profile which is inequivalent to the first compositenonlinearity profile by placing the sample and the reference materialproximate to one another in a second configuration with the sample linesubstantially collinear with the reference line. The method measures aFourier transform magnitude of the second composite nonlinearityprofile. The method calculates the sample nonlinearity profile using theFourier transform magnitudes of the sample nonlinearity profile, thereference nonlinearity profile, the first composite nonlinearityprofile, and the second composite nonlinearity profile.

In still another aspect of the present invention, a method measures anonlinearity profile of a sample. In accordance with the method, asample is provided that has at least one sample surface and having asample nonlinearity profile along a sample line through a predeterminedpoint on the sample surface. The sample line is oriented perpendicularlyto the sample surface. The method provides a reference material havingat least one reference surface and having a reference nonlinearityprofile along a reference line through a predetermined point on thereference surface. The reference line is oriented perpendicularly to thereference surface. The method forms a first composite sample having afirst composite nonlinearity profile by placing the sample and thereference material proximate to one another in a first configurationwith the sample line substantially collinear with the reference line.The method measures a Fourier transform magnitude of the first compositenonlinearity profile. The method forms a second composite sample havinga second composite nonlinearity profile which is inequivalent to thefirst composite nonlinearity profile by placing the sample and thereference material proximate to one another in a second configurationwith the sample line substantially collinear with the reference line.The method measures a Fourier transform magnitude of the secondcomposite nonlinearity profile. The method calculates the samplenonlinearity profile using the Fourier transform magnitudes of the firstcomposite nonlinearity profile and the second composite nonlinearityprofile.

In still another aspect of the present invention, a method measures asample temporal waveform of a sample optical pulse. In accordance withthe method, a sample optical pulse having a sample temporal waveform isprovided. The method measures a Fourier transform magnitude of thesample temporal waveform. The method provides a reference optical pulsehaving a reference temporal waveform. The method obtaines a Fouriertransform magnitude of the reference temporal waveform. The method formsa first composite optical pulse comprising the sample optical pulsefollowed by the reference optical pulse. The first composite opticalpulse has a first composite temporal waveform. The method measures aFourier transform magnitude of the first composite temporal waveform.The method provides a time-reversed pulse having a time-reversedtemporal waveform corresponding to the reference temporal waveform afterbeing time-reversed. The method forms a second composite optical pulsecomprising the sample optical pulse followed by the time-reversedoptical pulse. The method measures a Fourier transform of the secondcomposite temporal waveform. The method calculates the sample temporalwaveform using the Fourier transform magnitude of the sample temporalwaveform, the Fourier transform magnitude of the reference temporalwaveform, the Fourier transform magnitude of the first compositetemporal waveform, and the Fourier transform magnitude of the secondcomposite temporal waveform.

In still another aspect of the present invention, a method measures asample temporal waveform of a sample optical pulse. In accordance withthe method, a sample optical pulse having a sample temporal waveform isprovided. The method provides a reference optical pulse having areference temporal waveform. The method forms a composite optical pulsecomprising the sample optical pulse followed by the reference opticalpulse with a relative delay between the sample temporal waveform and thereference pulse waveform. The method measures a Fourier transformmagnitude squared of the composite optical pulse. The method calculatesan inverse Fourier transform of the measured Fourier transform magnitudesquared. The method calculates the sample temporal waveform using thecalculated inverse Fourier transform.

For purposes of summarizing the invention, certain aspects, advantagesand novel features of the invention have been described herein above. Itis to be understood, however, that not necessarily all such advantagesmay be achieved in accordance with any particular embodiment of theinvention. Thus, the invention may be embodied or carried out in amanner that achieves or optimizes one advantage or group of advantagesas taught herein without necessarily achieving other advantages as maybe taught or suggested herein.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A and 1B schematically illustrate the poling of a silica wafer.

FIG. 2 is a flow diagram of a method of measuring a second-order opticalnonlinearity profile of a sample in accordance with embodiments of thepresent invention.

FIG. 3A schematically illustrates a composite sample having an oddsecond-order optical nonlinearity profile.

FIGS. 3B and 3C schematically illustrate two measurement configurationsin accordance with embodiments of the present invention.

FIGS. 4A–4D schematically illustrate various embodiments of thesupplemental sample in accordance with embodiments of the presentinvention.

FIG. 5 illustrates exemplary Maker fringe (MF) profiles (in units of10⁻¹⁹ m²/W) measured from a common sample (shown as open circles) and acomposite sample (shown as crosses).

FIG. 6 illustrates the second-order nonlinearity profile d₃₃(z) (inunits of 10⁻¹² m/V) of the composite sample obtained by calculating theinverse Fourier transform of the Fourier transform of the compositenonlinearity profile.

FIG. 7 illustrates the original nonlinearity profile d₃₃(z) (in units of10⁻¹³ m/V) of the sample corresponding to the z>0 portion of thenonlinearity profile of FIG. 6.

FIG. 8 illustrates the comparison of the measured fringe profile (inunits of 10⁻¹⁹ m²/W) from the original sample (open circles) with atheoretical fringe profile calculated from d₃₃(z) (as shown in FIG. 7)obtained from the measured second harmonic (SH) signal from thecomposite sample (solid line).

FIG. 9 is a flow diagram of a method of measuring a sample nonlinearityprofile in accordance with embodiments of the present invention.

FIGS. 10A and 10B schematically illustrate single-pass configurationsfor measuring the Fourier transform magnitudes of the samplenonlinearity profile and the reference nonlinearity profile,respectively.

FIGS. 11A–11D schematically illustrate various configurations forforming composite samples in accordance with embodiments of the presentinvention.

FIGS. 12A and 12B schematically illustrate two odd nonlinearity profilesfor two arbitrary functions.

FIGS. 13A and 13B show arbitrarily selected nonlinearity profiles of thesample and the reference sample, respectively.

FIGS. 13C and 13D show the nonlinearity profiles of the two sandwichconfigurations, respectively, of the sample and reference sample ofFIGS. 13A and 13B.

FIG. 14A shows the calculated MF curves corresponding to the single-passconfiguration of the sample (shown as a solid line) and the referencesample (shown as a dashed line).

FIG. 14B shows the calculated MF curves corresponding to the double-passconfiguration of the first sandwich configuration (shown as a solidline) and the second sandwich configuration (shown as a dashed line).

FIGS. 15A and 15B respectively show the original nonlinearity profilesd_(A)(z) and d_(B)(z) as solid curves, and show the correspondingretrieved profiles shown as crosses.

FIG. 16 is a flow diagram of one embodiment of a method of measuring asample nonlinearity profile of a sample.

FIG. 17 is a flow diagram of one embodiment of a method for calculatingthe sample nonlinearity profile using the Fourier transform magnitudesof the first composite nonlinearity profile and the second compositenonlinearity profile.

FIGS. 18A-18D illustrate measured Maker fringe data curves (opencircles) and theoretical Maker fringe data curves (solid lines) from asample, a reference sample, a first composite sample, and a secondcomposite sample, respectively.

FIG. 19 illustrates the nonlinearity profiles d_(A)(z) and d_(B)(z) oftwo samples measured in accordance with embodiments of the presentinvention, and a nonlinearity profile d(z) measured using only onecomposite sample in accordance with embodiments of the presentinvention.

FIG. 20 is a flow diagram of a method in accordance with embodiments ofthe present invention for measuring physical functions.

FIG. 21 is a flow diagram of a method of determining the temporalwaveform of a laser pulse in accordance with embodiments of the presentinvention.

FIG. 22 schematically illustrates four-wave mixing (FWM) with pulsedpumps for providing the time-reversed pulse in accordance withembodiments of the present invention.

FIGS. 23A and 23B schematically illustrate femtosecond spectralholography for providing the time-reversed pulse in accordance withembodiments of the present invention.

FIG. 24 schematically illustrates one embodiment of a classicalintensity correlator utilizing a Michelson interferometer.

FIG. 25 schematically illustrates a general configuration for convertinga periodically repeated sequence of pulses into a periodically repeatedsequence of symmetric pulses using a movable phase conjugative mirror.

FIG. 26 illustrates an exemplary temporal waveform (in units of W/m²) ofan asymmetric input pulse compatible with embodiments of the presentinvention.

FIG. 27 illustrates the magnitude of the Fourier transform (unitless) ofthe autocorrelation function corresponding to the temporal waveform ofFIG. 26.

FIG. 28 illustrates the symmetric composite waveform (in units of W/m²)corresponding to the temporal waveform of FIG. 26.

FIG. 29 illustrates the magnitude of the Fourier transform (unitless) ofthe autocorrelation function of the symmetric composite waveform of FIG.28.

FIG. 30A illustrates the recovered symmetric temporal waveform (in unitsof W/m²).

FIG. 30B illustrates the difference (in units of W/m²) between therecovered symmetric temporal waveform and the temporal waveform of theoriginal pulse.

FIG. 31 schematically illustrates a system for another embodiment fordetermining the temporal waveform of a laser pulse.

FIGS. 32A and 32B illustrate the magnitude (in units of W/m²) and phaseof an arbitrary asymmetric complex envelope function to becharacterized.

FIG. 33 illustrates the intensity profile (in units of W/m²) for theasymmetric complex envelope function of FIG. 32A with the carrierfrequency oscillations.

FIG. 34 illustrates the symmetric temporal waveform (in units of W/m²)with the carrier frequency corresponding to the asymmetric complexenvelope function of FIG. 33.

FIGS. 35A and 35B illustrate the detected intensity (in units of W/m²)on the CCD imaging device for the symmetric pulse of FIG. 34, and thesingle pulse of FIG. 33, respectively.

FIGS. 36A and 36B illustrate the recovered symmetric temporal waveformand the original temporal waveform (in units of W/m²), respectively,including the carrier frequencies.

FIG. 37 illustrates the waveforms of FIGS. 36A and 36B (in units ofW/m²) overlaid with one another in an expanded time frame for comparisonpurposes.

FIG. 38 is a flow diagram of a method of measuring a sample temporalwaveform of a sample optical pulse in accordance with embodiments of thepresent invention.

FIG. 39 is a flow diagram of another method of measuring a sampletemporal waveform of a sample optical pulse in accordance withembodiments of the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

FIG. 2 is a flow diagram of a method 100 of measuring a second-orderoptical nonlinearity profile of a sample 10 of FIG. 1B in accordancewith embodiments of the present invention. While the flow diagramsherein illustrate particular embodiments with steps in a particularorder, other embodiments with different orders of steps are alsocompatible with the present invention.

In the method 100, the sample 10 is provided in an operational block110. The sample 10 has the second-order optical nonlinearity profile 15to be measured. In an operational block 120, a surface of the sample 10is placed proximate to a surface of a supplemental sample 20, asschematically illustrated in FIGS. 3A–3C and 4A–4D. As schematicallyillustrated in FIG. 3A, the sample 10 and supplemental sample 20 form afirst composite sample 30 having an odd second-order opticalnonlinearity profile 35. In an operational block 130, a Fouriertransform magnitude of the odd second-order optical nonlinearity profile35 of the composite sample 30 is measured. In an operational block 140,the sample nonlinearity profile is calculated using the Fouriertransform magnitude of the first composite nonlinearity profile.

In certain embodiments, calculating the sample nonlinearity profilecomprises using the measured Fourier transform magnitude to determine aFourier transform of the odd second-order optical nonlinearity profile35 of the composite sample 30, and calculating an inverse Fouriertransform of the determined Fourier transform. The calculated inverseFourier transform provides information regarding the second-orderoptical nonlinearity profile 15 of the sample 10. In certainembodiments, the information comprises the magnitude and sign of thesecond-order optical nonlinearity profile 15 as a function of depthbelow the surface of the sample 10. In other embodiments, theinformation comprises the position of the second-order opticalnonlinearity profile 15 below the surface of the sample 10.

In certain embodiments, the second-order optical nonlinearity profile 15of the sample 10 is non-symmetric, while in other embodiments, thesecond-order optical nonlinearity profile 15 of the sample 10 issymmetric about the origin (e.g., odd function of z). Other existingmethods are useful for measuring second-order optical nonlinearityprofiles which are symmetric about the origin, but such existing methodsare not compatible with non-symmetric profiles. Embodiments describedherein are compatible with both symmetric and non-symmetric second-orderoptical nonlinearity profiles. Since it is not generally known whetherthe second-order optical nonlinearity profile is non-symmetric orsymmetric about the origin prior to measurement, embodiments describedherein provide a general method of measurement independent of thesymmetry about the origin of the profile to be determined. In certainembodiments, the information comprises whether the second-order opticalnonlinearity profile is symmetric or non-symmetric about the origin.

Embodiments of the present invention can be used to determine thenonlinearity profile of any optically nonlinear material (e.g.,crystalline, amorphous, organic, or inorganic) in a bulk form or in afilm form. In certain embodiments, the nonlinear material comprises anorganic material, such as polymers and solids doped with dye molecules.In other embodiments, the nonlinear material comprises inorganicmaterials such as crystalline lithium niobate or amorphous materials(e.g., oxide-based, fluoride-based, or sulfide-based glasses).

In certain embodiments, the sample 10 comprises silica glass that hasbeen poled so as to induce a second-order optical nonlinearity profilein the sample 10. For example, a fused silica wafer comprised ofINFRASIL quartz glass and measuring 1″×1″×0.1″(i.e., 2.54 cm×2.54cm×2.54 mm) can be poled at approximately 270° C. in air by using ananode electrode placed proximate to one surface of the wafer (i.e., theanodic surface) and a cathode electrode placed proximate to the oppositesurface of the wafer (i.e., the cathodic surface) to apply approximately5 kV across the wafer for approximately 15 minutes. This procedureresults in a second-order optical nonlinearity profile approximately20–30 μm thick and localized beneath the anodic surface as described inmore detail by Thomas G. Alley et al. in Space Charge Dynamics inThermally Poled Fused Silica, Journal of Non-Crystalline Solids, Vol.242, 1998, pages 165–176, which is incorporated herein in its entirety.Other materials and objects (e.g., optical fibers) with inherent orinduced second-order optical nonlinearity profiles, and a wide range ofpoling conditions, are compatible with embodiments of the presentinvention.

FIGS. 3B and 3C schematically illustrate two measurement configurationsin accordance with embodiments of the present invention. In suchembodiments, the composite sample 30 has a surface normal direction n. Alaser 40 produces a laser beam 42 of light with a fundamental frequencyv. The laser beam 42 is incident on the composite sample 30 at an angleθ relative to the surface normal direction n. In the embodiment of FIG.3B, a detector 50 is positioned on the opposite side of the compositesample 30 so as to detect the second harmonic (SH) signal 52 (having afrequency 2v) from the composite sample 30. In certain embodiments,appropriate optical filters (not shown) are placed between the compositesample 30 and the detector 50 to reduce the laser power transmitted bythe composite sample 30 to be well below the weaker power level of theSH signal. The embodiment of FIG. 3B can be considered to be atransmission configuration. In the embodiment of FIG. 3C, the detector50 is positioned on the same side of the composite sample 30 as thelaser 40 so as to detect the SH signal 52. The embodiment of FIG. 3C canbe considered to be a reflection configuration. As used herein, the term“double pass” refers to configurations in which Maker fringe data areobtained from two wafers proximate to one another (e.g., as shown inFIGS. 3B and 3C). As used herein the term “single pass” refers toconfigurations in which Maker fringe data are obtained from a singlewafer.

In certain embodiments, the composite sample 30 is positioned so thatthe laser beam 42 is first incident on the sample 10, while in otherembodiments, the laser beam 42 is first incident on the supplementalsample 20. In certain embodiments, the SH signal 52 is measured as afunction of the incident angle θ of the laser beam 42 by rotating thecomposite sample 30 relative to the laser beam 42.

FIGS. 4A–4D schematically illustrate various embodiments of thesupplemental sample 20 in accordance with embodiments of the presentinvention. In certain embodiments, the supplemental sample 20 has asecond-order optical nonlinearity profile substantially identical tothat of the sample 10. In the embodiment schematically illustrated inFIG. 4A, the sample 10 and the supplemental sample 20 comprise twoportions of a common sample 60. For example, a fused silica wafer whichserves as the common sample 60 can be poled as described above,resulting in a poled region 62 beneath the anodic surface 63 of thewafer 60. The wafer can then be cut in half, producing two portions 64,65, which can serve as the sample 10 and the supplemental sample 20. Thesample 10 and the supplemental sample 20 can then be placed proximate toone another, thereby forming the composite sample 30.

By flipping the supplemental sample 20 180° to form the composite sample30, the second-order optical nonlinearity profile of the supplementalsample 20 is effectively multiplied by −1. The physical reason for thissign change is that during poling, the symmetry of the intrinsicmaterial is broken along the z direction. Thus, the second-order opticalnonlinearity profile of the composite sample 30 is an odd function. Incertain embodiments, the sample 10 and the supplemental sample 20 areplaced proximate to one another with the poled regions of the portions64, 65 proximate to one another (referred to herein as an anode-to-anodeconfiguration). In certain such embodiments, the two halves of thesurface 63 are in contact with one another, while in other embodiments,there is empty space or a spacer material between the two portions 64,65. This spacer material can comprise an index-matching gel whichreduces total internal reflection. In other embodiments, the sample 10and the supplemental sample 20 are placed proximate to one another withthe poled regions of the portions 64, 65 on the outer sides of thecomposite sample 30 (referred to herein as a cathode-to-cathodeconfiguration). In certain such embodiments, the two portions 64, 65 arein contact with one another, while in other embodiments, there is emptyspace or a spacer material (e.g., index-matching gel) between the twoportions 64, 65.

In the embodiment schematically illustrated in FIG. 4B, the supplementalsample 20 is prepared using substantially identical conditions as thoseused to prepare the sample 10. For example, two substantially identicalfused silica wafers 70, 74 can be poled sequentially or simultaneouslyas described above using substantially identical conditions, resultingin corresponding poled regions 71, 75 beneath corresponding surfaces 72,76. In this way, one wafer 70 serves as the sample 10, and the otherwafer 74 serves as the supplemental sample 20. The sample 10 and thesupplemental sample 20 can then be placed proximate to one another,thereby forming the composite sample 30. In certain embodiments, thesample 10 and the supplemental sample 20 are placed proximate to oneanother with the poled regions 71, 75 proximate to one another in theanode-to-anode configuration. In certain such embodiments, the twosurfaces 72, 76 are in contact with one another, while in otherembodiments, there is empty space or a spacer material (e.g.,index-matching gel), between the two wafers 70, 74. In otherembodiments, the sample 10 and the supplemental sample 20 are placedproximate to one another with the poled regions 71, 75 on opposite sidesof the composite sample 30 in the cathode-to-cathode configuration. Incertain such embodiments, the two wafers 70, 74 are in contact with oneanother, while in other embodiments, there is empty space or a spacermaterial (e.g., index-matching gel) between the two wafers 70, 74.

In the embodiment schematically illustrated in FIG. 4C, the supplementalsample 20 comprises a reflector 80 with a reflecting surface 81. Thesample 10 of such embodiments can comprise a wafer 82 with a poledregion 83 beneath a surface 84 of the wafer 82. The sample 10 and thesupplemental sample 20 can then be placed proximate to one another,thereby forming the composite sample 30. In certain embodiments, thesample 10 and the supplemental sample 20 are placed proximate to oneanother with the poled region 83 proximate to the reflecting surface 81.In certain such embodiments, the reflecting surface 81 and the surface84 are in contact with one another, while other embodiments have anempty space or a spacer material between the two surfaces 81, 84. Whendetecting the SH signal 52 in the reflection configuration, thereflector 80 of such embodiments provides an image of the second-ordernonlinearity profile of the sample 10 substantially identical to thesecond-order nonlinearity profile of the sample 10.

In alternative embodiments, as schematically illustrated in FIG. 4D, thesupplemental sample 20 and the poled sample 82 are placed proximate toone another with the reflecting surface 81 placed against the samplesurface 85 on the opposite side of the sample 10 from the surface 84. Incertain such embodiments, the two surfaces 81, 85 are in contact withone another, while in other embodiments, there is empty space or aspacer material between the two surfaces 81, 85.

In certain embodiments, placing the sample 10 and the supplementalsample 20 proximate to one another comprises sandwiching the sample 10and the supplemental sample 20 together. In certain such embodiments,the sample 10 and the supplemental sample 20 are clamped together, whilein other embodiments, the sample 10 and the supplemental sample 20 areglued together. Other methods of placing the sample 10 proximate to thesupplemental sample 20 are compatible with embodiments of the presentinvention.

FIG. 5 illustrates exemplary Maker fringe profiles measured from acommon sample 60 (shown as open circles) and from a composite sample 30(shown as crosses). As described above, the measured fringe profilecorresponds to the Fourier transform magnitude of the second-orderoptical nonlinearity profile of the composite sample 30. In certainembodiments, measuring the Fourier transform magnitude comprisesmeasuring the Maker fringe profile of the composite sample 30. Thecomposite sample 30 was formed by cutting the common sample 60 in halfand placing the two halves proximate to one another with the surfacesnear the poled regions in contact with one another. The fringe profileof the composite sample 30 is more intense and is shifted towards lowerangles than that of the common sample 60 because the nonlinear region ofthe composite sample 30 is twice as thick as the nonlinear region of thecommon sample 60.

FIG. 6 illustrates the second-order nonlinearity profile d₃₃(z) of thecomposite sample 30 obtained by calculating the inverse Fouriertransform of the Fourier transform calculated using the measured fringeprofile from the composite sample 30 as illustrated in FIG. 5. Theabscissa z=0 corresponds to the boundary between the mated surfaces ofthe sandwiched sample 10 and the supplemental sample 20. Byconstruction, the nonlinearity profile of the composite sample 30 is thejuxtaposition of the d₃₃(z) profile of the sample 10 and its mirrorimage with respect to the origin, i.e., −d₃₃(−z), from the supplementalsample 20. By retaining only the z>0 portion of the nonlinearity profileof FIG. 6, the original nonlinearity profile d₃₃(z) of the sample 10 isdirectly obtained, as illustrated in FIG. 7. The nonlinearity profile ofFIG. 7 represents an unambiguously derived nonlinearity profile of thethermally poled silica sample 10. The need for the phase information hasbeen eliminated by artificially creating an odd nonlinearity profilebased on the nonlinearity profile of the sample 10.

FIG. 7 also includes information regarding the depth location of thenonlinearity, which would not be available using prior methods, even ifthe phase information was retrieved. FIG. 7 shows that d₃₃(z) changessign and that its peak value is approximately 0.8 picometer per volt(pm/V=10⁻¹² m/V). This result is the highest reliable value of d₃₃(z)reported for thermally poled silica. The peak of the nonlinearityprofile is located approximately one micron under the anode surface, andthe poled region extends approximately 35 microns under the surface.

The main mechanism believed to be responsible for the second-orderoptical susceptibility χ⁽²⁾ in thermally poled silica is DCrectification of the third-order optical susceptibility χ⁽³⁾ of silica.As described more fully by D. Faccio et al. and T. G. Alley et al.(cited above), the second-order susceptibility χ⁽²⁾ is proportional toχ⁽³⁾E(z), where E(z) is the permanent electric field that developsinside the glass during poling.

The charge distribution within the glass which generates the permanentelectric field E(z) can be determined from the nonlinearity profiled₃₃(z). Decomposing the poled region into essentially thin infiniteplanes, the electric field E(z) is related to the charge density ρ(z) bythe one-dimensional form of Maxwell's equation, δE/δz=ρ(z)/ε, where ε isthe dielectric susceptibility of the medium. Since d₃₃(z) isproportional to E(z), the charge distribution ρ(z) can be derived bydifferentiating the profile d₃₃(z).

The nonlinearity profile derived in accordance with embodiments of thepresent invention can be independently verified by comparing themeasured Maker fringe profile from the original sample 10 (shown as opencircles in FIG. 5 and FIG. 8) with the theoretical fringe profile (solidcurve in FIG. 8) calculated from d₃₃(z) obtained from the measured SHsignal from the composite sample 30 (FIG. 7). This comparison,illustrated in FIG. 8, shows that the two fringe profiles agreereasonably well, and that in particular, the spatial uniformity of thesecond-order susceptibility of the sample 10 was sufficient to infer thenonlinearity profile reliably.

Embodiments of the present invention enable the second-ordernonlinearity profile of an optically nonlinear film to be inferredunambiguously from a Maker fringe profile measurement. As describedabove, the nonlinearity profile of an exemplary thermally poled silicasample has been determined to: (i) have a peak value of approximately0.8 pm/V, (ii) extend approximately 35 microns below the sample surface,and (iii) take both positive and negative values. Such magnitude andspatial information of the nonlinearity profile and of the chargedistribution has significant implications in the design of futuredevices based on thermally poled silica.

Embodiments described above can be considered to be special cases of amore general method of determining the second-order nonlinearityprofile. In the embodiments described above in which the sample 10 hasbeen cut into two pieces which form the composite sample 30, twoassumptions have been made: (1) the nonlinearity profiles of both thesample 10 and the supplemental sample 20 have the same functionaldependence normal to the anode surface (i.e., ƒ(z)); and (2) both thesample 10 and the supplemental sample 20 have the same nonlinearstrength normal to the anode surface (i.e., d₃₃(z)=K·ƒ(z), where K isthe same constant for both the sample 10 and the supplemental sample20). These assumptions in principle limit the application ofabove-described embodiments to nonuniform poled samples.

In another embodiment, the nonlinearity profile of a sample is uniquelydetermined without either of the two assumptions described above. Inaddition, the sample 10 need not be cut into two pieces to determine thenonlinearity profile.

FIG. 9 is a flow diagram of a method 150 of measuring a samplenonlinearity profile 15 of a sample 10. In an operational block 152, asample 10 having at least one sample surface and having a samplenonlinearity profile 15 along a sample line through a predeterminedpoint on the sample surface is provided. The sample line is orientedperpendicularly to the sample surface. In an operational block 154, aFourier transform magnitude of the sample nonlinearity profile 15 ismeasured. In an operational block 156, a reference material having atleast one reference surface and having a reference nonlinearity profilealong a reference line through a predetermined point on the referencesurface is provided. The reference line is oriented perpendicularly tothe reference surface. In an operational block 158, a Fourier transformmagnitude of the reference nonlinearity profile is obtained. In anoperational block 160, a first composite sample having a first compositenonlinearity profile is formed. The first composite sample is formed byplacing the sample 10 and the reference material proximate to oneanother in a first configuration with the sample line substantiallycollinear with the reference line. In an operational block 162, aFourier transform magnitude of the first composite nonlinearity profileis measured. In an operational block 164, a second composite samplehaving a second composite nonlinearity profile inequivalent to the firstcomposite nonlinearity profile is formed. The second composite sample isformed by placing the sample 10 and the reference material proximate toone another in a second configuration with the sample line substantiallycollinear with the reference line. In an operational block 166, aFourier transform magnitude of the second composite nonlinearity profileis measured. In an operational block 168, the sample nonlinearityprofile 15 is calculated using the Fourier transform magnitudes of thesample nonlinearity profile, the reference nonlinearity profile, thefirst composite nonlinearity profile, and the second compositenonlinearity profile.

In certain embodiments, the Fourier transform magnitude of the samplenonlinearity profile is measured in the operational block 154 bymeasuring the Maker fringe (MF) data of the sample in a single-passconfiguration as schematically illustrated by FIG. 10A. The MF data ofthe sample (MF₁) is thus representative of the Fourier transformmagnitude of the sample nonlinearity profile along a sample line througha predetermined point (labelled “A” in FIG. 10A) on the sample surface.The sample line is oriented perpendicularly to the sample surface.Similarly, as schematically illustrated in FIG. 10B, the Fouriertransform magnitude of the reference nonlinearity profile can bemeasured by measuring the MF data of the reference material in asingle-pass configuration. The MF data of the reference material (MF₂)is thus representative of the Fourier transform magnitude of thereference nonlinearity profile along a reference line through apredetermined point (labelled “B” in FIG. 10B) on the reference surface.As used below, the MF data of the sample is expressed as:MF ₁(ƒ)=|D _(A)(ƒ)|²  (2)and the MF data of the reference sample is expressed as:MF ₂(ƒ)=|D _(B)(ƒ)|²  (3)where the Fourier transform of the sample nonlinearity profile d_(A)(z)is denoted by

${{{d_{A}(z)}\;\overset{FT}{\longrightarrow}{D_{A}(f)}} = {{{D_{A}(f)}}{\mathbb{e}}^{{j\phi}_{A}{(f)}}}},$the Fourier transform of the reference nonlinearity profile d_(B)(z) isdenoted by

${{{d_{B}(z)}\;\overset{FT}{\longrightarrow}{D_{B}(f)}} = {{{D_{B}(f)}}{\mathbb{e}}^{{j\phi}_{B}{(f)}}}},$and ƒ is the spatial frequency. The spatial frequency ƒ is given by

${f = {\pm \left| {2\frac{{n_{1}\cos\;\theta_{\omega}} - {n_{2}\cos\;\theta_{2\;\omega}}}{\lambda}} \right|}},$where λ is the fundamental wavelength, and n₁, n₂, θ_(ω), and θ_(2ω)arethe refractive indices and internal propagation angles at thefundamental and second harmonic wavelengths, respectively.

In general, d_(A)(z) does not equal d_(B)(z). For both d_(A)(z) andd_(B)(z), the poled region is assumed to be in the z≦0 half of thez-coordinate system where z=0 defines the anodic surfaces. This choiceof coordinate system ensures that for z>0, d_(A)(z)=d_(B)(z)=0. Inaddition, the depth of the poled region at the sample surface is W_(A)such that d_(A)(z)=0 for z<−W_(A), and the depth of the poled region atthe reference surface is W_(B) such that d_(B)(z)=0 for z<−W_(B).

FIGS. 11A–11D schematically illustrate various configurations forforming composite samples in accordance with embodiments of the presentinvention. Each of the composite samples of FIGS. 11A–11D has acorresponding composite nonlinearity profile. In FIG. 11A, the compositesample is formed by placing the anodic surface of the sample and theanodic surface of the reference material proximate to one another, andis referred to herein as an anode-to-anode configuration. In FIG. 11B,the composite sample is formed by placing the cathodic surface of thesample and the cathodic surface of the reference material proximate toone another, and is referred to herein as a cathode-to-cathodeconfiguration. In FIG. 11C, the composite sample is formed by placingthe anodic surface of the sample and the cathodic surface of thereference material proximate to one another, and is referred to hereinas an anode-to-cathode configuration. In FIG. 11D, the composite sampleis formed by placing the cathodic surface of the sample and the anodicsurface of the reference material proximate to one another, and isreferred to herein as a cathode-to-anode configuration.

In certain embodiments, the Fourier transform magnitude of the firstcomposite nonlinearity profile is measured in the operational block 162(FIG. 9) by measuring the MF data of the first composite sample in adouble-pass configuration. In embodiments in which the first compositesample has a first configuration as schematically illustrated by one ofFIGS. 11A–11D, the MF data of the first composite sample is thusrepresentative of the Fourier transform magnitude of the first compositenonlinearity profile along the dashed line of the corresponding one ofFIGS. 11A–11D.

In certain embodiments, the Fourier transform magnitude of the secondcomposite nonlinearity profile is measured in the operational block 166(FIG. 9) by measuring the MF data of the second composite sample in adouble-pass configuration. The configuration of the second compositesample is chosen to provide a second composite nonlinearity profileinequivalent to the first composite nonlinearity profile. For example,if the first configuration of the first composite sample is that of FIG.11A or FIG. 11B, then the second configuration of the second compositesample can be that of FIG. 11C or FIG. 11D. Similarly, if the firstconfiguration of the first composite sample is that of FIG. 11C or FIG.11D, then the second configuration of the second composite sample can bethat of FIG. 11A or FIG. 11B. The MF data of the second composite sampleis thus representative of the Fourier transform magnitude of the secondcomposite nonlinearity profile along the dashed line of thecorresponding one of FIGS. 11A–11D.

The MF data of the anode-to-anode configuration of FIG. 11A (MF_(S1))contains the same information as the MF data of the cathode-to-cathodeconfiguration of FIG. 11B (MF_(S2)). Similarly, the MF data of theanode-to-cathode configuration of FIG. 11C (MF_(S3)) contains the sameinformation as the MF data of the cathode-to-anode configuration of FIG.11D (MF_(S4)). The MF data corresponding to the four possibleconfigurations shown in FIGS. 11A–11D can be written as:MF _(S1) =|D _(A)|² +|D _(B)|²−2|D _(A) ∥D _(B)|cos(φ_(A)+φ_(B))  (4)MF _(S2) =|D _(A)|² +|D _(B)|²−2|D _(A) ∥D_(B)|cos(φ_(A)+φ_(B)+2φ₀)  (5)MF _(S3) =|D _(A)|² +|D _(B)|²+2|D _(A) ∥D _(B)|cos(φ_(A)−φ_(B)+φ₀)  (6)MF _(S4) =|D _(A)|² +|D _(B)|²+2|D _(A) ∥D _(B)|cos(φ_(A)−φ_(B)−φ₀)  (7)The dependence of all these quantities on the spatial frequency has beenomitted in Equations 4–7 for clarity.

From Equations 4 and 5, it can be seen that MF_(S1) and MF_(S2) areequivalent to one another (i.e., they have the same information in termsof φ_(A) and φ_(B)). The extra modulation term φ₀=2πfL (where L is thesample thickness) does not contribute information regarding thenonlinearity profiles since it corresponds to a modulation term due tothe summed thicknesses of the sample and reference material. In Equation5, the factor of 2 in front of φ₀ comes from the assumption that thethickness of the sample and of the reference material are the both equalto L. Similarly, MF_(S3) and MF_(S4) are equivalent to one another, butare inequivalent to MF_(S1) and MF_(S2). Thus, in certain embodiments,either MF_(S1) or MF_(S2) is used as a first independent source ofinformation, and either MF_(S3) or MF_(S4) is used as a secondindependent source of information. While the description below focuseson the configurations of FIGS. 11A and 11C, in certain embodiments, oneor both of these configurations can be substituted by its equivalentconfiguration in FIGS. 11B and 11D, respectively. For embodiments inwhich the sample and the reference material are spaced from one another(e.g., by an index-matching gel), Equations 4–7 can be modified toreflect the additional phase φ₁(f)=2πfL_(G) where L_(G) is the thicknessof the space between the sample and the reference material.

For configurations of FIGS. 11A and 11C, the effective nonlinearityprofiles can respectively be written as d^(S1)(z)=d_(A)(z)−d_(B)(−z) andd^(S2)(z)=d_(A)(z)+d_(B)(z−L), where S1 and S2 respectively denote theanode-to-anode and anode-to-cathode configurations of FIGS. 11A and 11C.As used herein, the z=0 point is assumed to be at the anodic surface ofthe sample (i.e., at the boundary between the sample and the referencematerial), and L is the total thickness of the reference sample whereL≧W_(B). Also, as used herein, the Fourier transforms of these functionsare denoted as

${d^{S1}(z)}\;\overset{FT}{\longrightarrow}{D^{S1}(f)}$and

${{d^{S2}(z)}\;\overset{FT}{\longrightarrow}{D^{S2}(f)}}.$For embodiments in which there is a space with a thickness LG betweenthe sample and the reference material, the effective nonlinearityprofiles can be respectively written asd^(S1)(z)=d_(A)(z)−d_(B)(−z+L_(G)) and d^(S2)(z)=d_(A)(z)+d_(B)(z−L),where L=L_(B)+L_(G) and L_(B) is the thickness of sample B.

The Fourier transform magnitudes of the sample nonlinearity profile, thereference nonlinearity profile, the first composite nonlinearityprofile, and the second composite nonlinearity profile are used tocalculate the sample nonlinearity profile in the operational block 168.The MF measurements corresponding to the two composite samples of FIGS.11A and 11C can be expressed as:MF _(S1) =|D ^(S1)(ƒ)|² =D _(A)(ƒ)−D _(B)(−ƒ) |² =||D _(A)(ƒ)|e^(jφ)^(A) ^((ƒ)−|D) _(B)(ƒ)|e^(−jφ) ^(B) ^((ƒ))|²  (8)MF _(S2) =|D ^(S2)(ƒ)|² =D _(A)(ƒ)+D _(B)(ƒ)·e^(−jπƒL)|² =|D_(A)(ƒ)|·e^(jφ) ^(A) ^((ƒ)) +|D _(B)(ƒ)·e^(j[φ) ^(B) ^((ƒ)−φ) ⁰ ^((ƒ)]|)²  (9)where φ₀(ƒ)=2πƒL. Expanding the absolute value sign in Equations 8 and9, MF_(S1 and MF) _(S2) can be expressed as:MF _(S1) =|D _(A) ^(|2) +|D _(B) ^(|2)−2|D _(A) ||D _(B)|cos(φ_(A)+φ_(B))  (10)MF _(S2) =|D _(A) ^(|2) +|D _(B) ^(|2)+2|D _(A) ||D _(B)|cos(φ_(A)−φ_(B)+φ₀)  (11)where the frequency dependencies of all the functions have been droppedfor convenience.

In certain embodiments, once the four sets of MF data (MF₁, MF₂, MF_(S1)and MF_(S2), given by Equations 2, 3, 10, and 11, respectively) areeither measured or otherwise obtained, they can be used to express thefollowing quantities:

$\begin{matrix}{{\cos\left( {\phi_{A} + \phi_{B}} \right)} = {\frac{{- {MF}_{S1}} + {MF}_{1} + {MF}_{2}}{2\sqrt{{MF}_{1} \cdot {MF}_{2}}} = \frac{\alpha}{\Delta}}} & (12) \\{{\cos\left( {\phi_{A} - \phi_{B} + \phi_{0}} \right)} = {\frac{{MF}_{S2} - {MF}_{1} - {MF}_{2}}{2\sqrt{{MF}_{1} \cdot {MF}_{2}}} = \frac{\beta}{\Delta}}} & (13)\end{matrix}$Note that α, β, and Δ are functions of frequency, ƒ, and are fullydetermined by the MF data from the sample, reference material, and thefirst and second composite samples. Equations 12 and 13 can be rewrittenin the following form:

$\begin{matrix}{{\phi_{A} + \phi_{B}} = \left. {{2{\pi \cdot m}} \pm} \middle| {\cos^{- 1}\left( \frac{\alpha}{\Delta} \right)} \right.} & (14) \\{{\phi_{A} - \phi_{B}} = \left. {{2{\pi \cdot n}} \pm} \middle| {\cos^{- 1}\left( \frac{\beta}{\Delta} \right)} \middle| {- \phi_{0}} \right.} & (15)\end{matrix}$where m and n can take any integer value (0,±1,±2, . . . ). Note alsothat the output of the inverse cosine function is between 0 and π.Equations 14 and 15 can be combined to express the following quantities:

$\begin{matrix}{{\phi_{A} + \frac{\phi_{0}}{2}} = \left. {{\pi \cdot k} \pm \frac{1}{2}} \middle| {\cos^{- 1}\left( \frac{\alpha}{\Delta} \right)} \middle| {\pm \frac{1}{2}} \middle| {\cos^{- 1}\left( \frac{\beta}{\Delta} \right)} \right|} & (16) \\{{\phi_{B} - \frac{\phi_{0}}{2}} = \left. {{\pi \cdot l} \pm \frac{1}{2}} \middle| {\cos^{- 1}\left( \frac{\alpha}{\Delta} \right)} \middle| {\mp \frac{1}{2}} \middle| {\cos^{- 1}\left( \frac{\beta}{\Delta} \right)} \right|} & (17)\end{matrix}$where k and 1 are any integers. By taking the cosine of both sides andtaking their absolute values, Equations 16 and 17 can be rewritten inthe following form:

$\begin{matrix}{\left| {\cos\left( {\phi_{A} + \frac{\phi_{0}}{2}} \right)} \right| = \left| {\cos\left( \left. \frac{1}{2} \middle| {\cos^{- 1}\left( \frac{\alpha}{\Delta} \right)} \middle| {\pm \frac{1}{2}} \middle| {\cos^{- 1}\left( \frac{\beta}{\Delta} \right)} \right| \right)} \right|} & (18) \\{\left| {\cos\left( {\phi_{B} - \frac{\phi_{0}}{2}} \right)} \right| = \left| {\cos\left( \left. \frac{1}{2} \middle| {\cos^{- 1}\left( \frac{\alpha}{\Delta} \right)} \middle| {\mp \frac{1}{2}} \middle| {\cos^{- 1}\left( \frac{\beta}{\Delta} \right)} \right| \right)} \right|} & (19)\end{matrix}$Equations 18 and 19 provide useful information towards the calculationof the sample nonlinearity profile. Note that the right-hand sides ofboth Equations 18 and 19 have the same two possible values, which aregiven by:

$\begin{matrix}{P_{1} = \left| {\cos\left( \left. \frac{1}{2} \middle| {\cos^{- 1}\left( \frac{\alpha}{\Delta} \right)} \middle| {+ \frac{1}{2}} \middle| {\cos^{- 1}\left( \frac{\beta}{\Delta} \right)} \right| \right)} \right|} & (20) \\{P_{2} = \left| {\cos\left( \left. \frac{1}{2} \middle| {\cos^{- 1}\left( \frac{\alpha}{\Delta} \right)} \middle| {- \frac{1}{2}} \middle| {\cos^{- 1}\left( \frac{\beta}{\Delta} \right)} \right| \right)} \right|} & (21)\end{matrix}$where P₁ and P₂ denote these two possible values. Notice also that if

${\left| {\cos\left( {\phi_{A} + \frac{\phi_{0}}{2}} \right)} \right| = P_{1}},$then

$\left| {\cos\left( {\phi_{B} - \frac{\phi_{0}}{2}} \right)} \right| = {P_{2}.}$The reverse is also true, i.e., if

${\left| {\cos\left( {\phi_{A} + \frac{\phi_{0}}{2}} \right)} \right| = P_{2}},$then

$\left| {\cos\left( {\phi_{B} - \frac{\phi_{0}}{2}} \right)} \right| = {P_{1}.}$

Using the Fourier transforms of the sample and reference materialnonlinearity profiles, the following quantities can be expressed:

$\begin{matrix}{{d_{A}\left( {z + \frac{L}{2}} \right)} + {{{d_{A}\left( {{- z} + \frac{L}{2}} \right)}\;\overset{\mspace{40mu}{FT}\mspace{45mu}}{\longrightarrow}2}{D_{A}}{\cos\left( {\phi_{A} + \frac{\phi_{0}}{2}} \right)}}} & (22) \\{{d_{B}\left( {z - \frac{L}{2}} \right)} + {{{d_{B}\left( {{- z} - \frac{L}{2}} \right)}\;\overset{\mspace{40mu}{FT}\mspace{45mu}}{\longrightarrow}2}{D_{B}}{\cos\left( {\phi_{B} - \frac{\phi_{0}}{2}} \right)}}} & (23)\end{matrix}$FIGS. 12A and 12B schematically illustrate two symmetric nonlinearityprofiles for two arbitrary functions d_(A)(z) and d_(B)(z). In FIGS. 12Aand 12B, the sharp boundaries at z=±L/2 correspond to the anodicsurfaces of the sample and reference material. The major differencebetween FIGS. 12A and 12B is that the nonzero nonlinearity profile is inthe

${z} < \frac{L}{2}$region for

${{d_{B}\left( {z - \frac{L}{2}} \right)} + {d_{B}\left( {{- z} - \frac{L}{2}} \right)}},$whereas for

${{d_{A}\left( {z + \frac{L}{2}} \right)} + {d_{A}\left( {{- z} + \frac{L}{2}} \right)}},$the nonzero nonlinearity profile is in the

${z} > \frac{L}{2}$region.

Equations 22 and 23 are connected to Equations 18 and 19 because of the

$\left| {\cos\left( {\phi_{A} - \frac{\phi_{0}}{2}} \right)} \right|$and the

$\left| {\cos\left( {\phi_{B} + \frac{\phi_{0}}{2}} \right)} \right|$terms. This connection can be used to remove the final ambiguity ofwhether

$\left| {\cos\left( {\phi_{A} - \frac{\phi_{0}}{2}} \right)} \right| = {\left. {P_{1}\mspace{14mu}{or}}\mspace{14mu} \middle| {\cos\left( {\phi_{A} - \frac{\phi_{0}}{2}} \right)} \right| = {P_{2}.}}$Since

${d_{A}\left( {z + \frac{L}{2}} \right)} + {d_{A}\left( {{- z} + \frac{L}{2}} \right)}$is a symmetric and real function, its Fourier transform magnitude issufficient to uniquely recover it. The same is also true for

${{d_{B}\left( {z - \frac{L}{2}} \right)} + {d_{B}\left( {{- z} - \frac{L}{2}} \right)}},$which is also a symmetric and real function.

Note that for a real and symmetric function, the Hartley transform isthe same as the Fourier transform. The final ambiguity can then beremoved using the Hartley transform, since for a real and compactsupport function (i.e., one that equals zero outside a finite region),the intensity of the Hartley transform is enough to uniquely recover theoriginal function. See, e.g., N. Nakajima in Reconstruction of a realfunction from its Hartley-transform intensity, Journal of the OpticalSociety of America A, Vol. 5, 1988, pages 858–863, and R. P. Millane inAnalytic Properties of the Hartley Transform and their Implications,Proceedings of the IEEE, Vol. 82, 1994, pages 413–428, both of which areincorporated in their entirety by reference herein.

In certain embodiments, the ambiguity is removed in the followingmanner. If

${\left| {\cos\left( {\phi_{A} + \frac{\phi_{0}}{2}} \right)} \right| = P_{1}},$then the function 2|D_(A)|P₁ is the Fourier transform magnitude

${d_{A}\left( {z + \frac{L}{2}} \right)} + {d_{A}\left( {{- z} + \frac{L}{2}} \right)}$(see Equation 22). For real and symmetric functions, the Hartleytransform can be used to uniquely recover the original function, asdescribed by Nakajima and by Millane, as referenced above. If theinverse Fourier transform of the Fourier transform obtained from2|D_(A)|P₁ (i.e., the Fourier transform magnitude of

$\left( {{i.e.},\;{{{the}\mspace{14mu}{Fourier}\mspace{14mu}{transform}\mspace{14mu}{magnitude}\mspace{14mu}{of}\mspace{14mu}{d_{A}\left( {z + \frac{L}{2}} \right)}} + {d_{A}\left( {{- z} + \frac{L}{2}} \right)}}} \right)$gives the poled region in

${{z} > \frac{L}{2}},$then

$\left| {\cos\left( {\phi_{A} + \frac{\phi_{0}}{2}} \right)} \right| = {P_{1}.}$Otherwise,

$\left| {\cos\left( {\phi_{A} + \frac{\phi_{0}}{2}} \right)} \right| = {P_{2}.}$This result can be double-checked by computing the inverse Fouriertransform of the Fourier transform obtained from 2|D_(A)|P₂ andconfirming that the poled region is given by

${z} > {\frac{L}{2}.}$Note finally that the inverse Fourier transforms of both Fouriertransforms obtained from the Fourier transform magnitudes 2|D_(A)|P₁ and2|D_(A)|P2 do not at the same time give a poled region in

${{z} > \frac{L}{2}},$because either 2|D_(B)|P₁ or 2|D_(B)|P₂ has to y a poled region in

${z} < \frac{L}{2}$for the function

${d_{B}\left( {z + \frac{L}{2}} \right)} + {d_{B}\left( {{- z} + \frac{L}{2}} \right)}$(see Equation 23).

In certain other embodiments, the ambiguity is removed by computing theinverse Fourier transform of the Fourier transforms obtained from both2|D_(A)|P_(I and) 2|D_(A)|P2. The results should yield two symmetricfunctions as shown in FIGS. 12A and 12B. Taking only the z>0 portion ofthe resulting profiles yields two alternative profiles for d_(A)(z). Thesame procedure can be applied for d_(B)(z). By computing the theoreticalMF curve of these two possible d_(A)(z) profiles and comparing theresults with the measured MF₁(ƒ)=|D_(A)(ƒ)|² data, it is straightforwardto choose the correct possibility.

In an exemplary embodiment, FIGS. 13A and 13B show arbitrarily selectednonlinearity profiles of the sample and the reference sample,respectively, and FIGS. 13C and 13D show the nonlinearity profiles ofthe two sandwich configurations, respectively.

FIG. 14A shows the calculated MF curves corresponding to the single-passconfiguration of the sample (shown as a solid line) and the referencesample (shown as a dashed line). FIG. 14B shows the calculated MF curvescorresponding to the double-pass configuration of the first sandwichconfiguration (shown as a solid line) and of the second sandwichconfiguration (shown as a dashed line).

FIGS. 15A and 15B respectively show the original nonlinearity profilesd_(A)(z) and d_(B)(z) as solid curves, and show the correspondingprofiles retrieved using the above-described method as crosses.

The foregoing description includes ways to uniquely recover the twoarbitrary nonlinearity profiles (d_(A)(z) and d_(B)(z)) using four setsof MF data. This result is significant since (1) the technique isapplicable to even nonuniform poled samples; and (2) the sample does notneed to be cut into two halves. The technique utilizes a referencematerial with known MF data. In certain embodiments, obtaining the MFdata of the reference material comprises measuring the MF data. In otherembodiments, the MF data of the reference material is previouslymeasured and stored in memory, and obtaining the MF data comprisesreading the MF data from memory. Embodiments in which the MF data ispreviously stored are preferable, since the number of MF measurementswill be reduced from 4 to 3. Note that it is not necessary to know thereference nonlinearity profile for this technique to work, since onlyits corresponding MF data is used. In addition, a common referencesample can be used to characterize a plurality of samples.

In an exemplary embodiment, the sample and reference material are cutfrom the same poled wafer, and the MF data from the sample and referencematerial (given by Equations 2 and 3, respectively) are related by aconstant factor (i.e., MF₁=C·MF₂). Since points A and B are from thesame poled surface, it can be assumed that:d _(A)(z)=√{square root over (C)}·d_(B)(z)  (24)It follows from Equation (24) that |D_(A)|=√{square root over(C)}·|D_(B)| and φ_(A)=φ_(B). In this exemplary embodiment, the MF dataof the second composite sample (MF_(S2), given by Equation 11) is notrequired to determine the sample nonlinearity profile. This result canbe seen by inserting φ_(A)=φ_(B) into Equation 11 and observing that allthe phase information related to the sample nonlinearity profile isremoved from Equation 11. Such embodiments are preferable because themathematics of the solution is significantly more simple and is obtainedwith an experimental simplicity. In this preferred embodiment, themathematical derivation stops at Equation 12, thereby avoiding thesubsequent equations.

Using the fact that |D_(A)|=√{square root over (C)}·|D_(B)|andφ_(A)=φ_(B), Equation 12 can be rewritten as:

$\begin{matrix}{{2 \cdot {D_{A}} \cdot {{\cos\left( \phi_{A} \right)}}} = \sqrt{\frac{{{MF}_{1} \cdot \left( {1 + \sqrt{C}} \right)^{2}} - {C \cdot {MF}_{S1}}}{\sqrt{C}}}} & (25)\end{matrix}$

The left-hand-side of Equation 25 is the Fourier transform magnitude ofd_(A)(z)+d_(A)(−z) and for such real and symmetric functions, theFourier transform magnitude is sufficient to uniquely recover theoriginal profile. The unique recovery of the sample nonlinearity profiled_(A)(z)=√{square root over (C)}·d_(B)(z) is thus achieved usingEquation 25. Note that embodiments in which C=1 yield the same solutionas do embodiments in accordance with the method of FIG. 2. This resultcan be verified by putting C=1 in Equation 25. Note that for C=1, andusing Equation 2, MF_(S1)=4|D_(A) ^(|2) sin²(φ_(A)).

As described above, by flipping a nonlinear poled sample 180° to matewith the anode surface of another sample, the nonlinear profile of theflipped sample d(−z)changes sign to −d(z). Thus, the nonlinearityprofile of the composite sample equals d(z)−d(−z). The effect of thissign change of the flipped sample is that the nonlinearity profile ofthe composite sample is now an odd function, i.e., it is symmetric aboutthe origin. For such an odd and real function, the Fourier transform ispurely imaginary and odd, and the phase of the Fourier transform isequal to ±π/2. Measurements of the MF curve of the composite sampleprovide the square of the Fourier transform magnitude of thenonlinearity profile d(z)−d(−z), i.e., MF_(S1)=4|D_(A) ⁵¹ ² sin²(φ_(A)). The Fourier transform of d(z)−d(−z) 2|D_(A)|sin (φ_(A)).Therefore, the MF measurement of the composite sample is equivalent tomeasuring the Fourier transform magnitude of the nonlinear profiled(z)−d(−z). But for real and odd functions, the Fourier transformmagnitude is the same as the Hartley transform magnitude. Thus, the MFmeasurement provides a measurement of the Hartley transform magnitude ofthe real and odd function, i.e., the nonlinear profile of the compositesample dz)−d(−z). The retrieval of a real function from only its Hartleytransform magnitude can be performed in various ways, as described byNakajima and by Millane, as referenced above.

Certain embodiments described above yield the sample nonlinearityprofile as well as the reference nonlinearitly profile. In suchembodiments, the reference material can comprise a second sample with asecond sample nonlinearity profile to be measured. Thus, thenonlinearity profiles of two samples can be measured concurrently.

In certain embodiments, the same reference material can be used formeasuring the nonlinearity profiles of a plurality of samples. If thesame reference material is used with different samples, the measuredreference nonlinearity profile should be substantially the same fromeach of the measurements. Comparison of the measured referencenonlinearity profiles accompanying each of the plurality of measuredsample nonlinearity profiles then provides an indication of theconsistency of the measurements across the plurality of samples.

In certain embodiments, the sample nonlinearity profile can becalculated using a more generalized and flexible procedure. FIG. 16 is aflow diagram of one embodiment of a method 170 of measuring a samplenonlinearity profile 15 of a sample 10. In an operational block 171, asample 10 having at least one sample surface and having a samplenonlinearity profile along a sample line through a predetermined pointon the sample surface is provided. The sample line is orientedperpendicularly to the sample surface. In an operational block 172, areference material having at least one reference surface and having areference nonlinearity profile along a reference line through apredetermined point on the reference surface is provided. The referenceline is oriented perpendicularly to the reference surface. In anoperational block 173, a first composite sample having a first compositenonlinearity profile is formed. The first composite sample is formed byplacing the sample 10 and the reference material proximate to oneanother in a first configuration with the sample line substantiallycollinear with the reference line. In an operational block 174, aFourier transform magnitude of the first composite nonlinearity profileis measured. In an operational block 175, a second composite samplehaving a second composite nonlinearity profile inequivalent to the firstcomposite nonlinearity profile is formed. The second composite sample isformed by placing the sample 10 and the reference material proximate toone another in a second configuration with the sample line substantiallycollinear with the reference line. In an operational block 176, aFourier transform magnitude of the second composite nonlinearity profileis measured. In an operational block 177, the sample nonlinearityprofile 15 is calculated using the Fourier transform magnitudes of thefirst composite nonlinearity profile and the second compositenonlinearity profile.

FIG. 17 is a flow diagram of one embodiment of the operational block 177for calculating the sample nonlinearity profile 15. In an operationalblock 178, an inverse Fourier transform of the difference between theFourier transform magnitudes of the first and second compositenonlinearity profiles is calculated. In an operational block 179, theinverse Fourier transform is separated into a first convolution functionand a second convolution function. In an operational block 180, theFourier transform of the first convolution function and the Fouriertransform of the second convolution function are calculated. In anoperational block 181, the phase and amplitude of the Fourier transformof the sample nonlinearity profile are calculated using the Fouriertransforms of the first and second convolution functions. In anoperational block 182, the inverse Fourier transform of the Fouriertransform of the sample nonlinearity profile 15 is calculated.

In certain embodiments, once the two sets of MF data (MF_(S1) andMF_(S2), given by Equations 10 and 11, respectively) are measured, thenonlinearity profile of the sample can be computed in the followingmanner. The nonlinear coefficient profiles of sample A and B are definedas d_(A)(z) and d_(B)(z), respectively, where z is in the directionperpendicular to the sample (see FIGS. 13A and 13B). The respectivethicknesses of the nonlinear regions in samples A and B are referred toas W_(A) and W_(B), respectively. By definition, the nonlinear regionsare confined to z≦0 (i.e., for z>0, d_(A)(z)=d_(B)(z)=0). Classical MFmeasurements performed on sample A and sample B alone would yield, withsome known proportionality constant, the square of the Fourier transformmagnitude of d_(A)(z) and d_(B)(z):MF _(A)(ƒ)=|D _(A)(ƒ)|²  (26)MF _(B)(ƒ)=|D _(B)(ƒ)|²  (27)where |D_(A)(ƒ)| and |D_(B)(ƒ)| are the Fourier transform magnitudes ofd_(A)(z) and d_(B)(z), respectively,

$f = {\pm {{2\frac{{n_{1}\cos\;\theta_{\omega}} - {n_{2}\cos\;\theta_{2\;\omega}}}{\lambda}}}}$is the spatial frequency, where λ is laser (fundamental) wavelength, andn₁, n₂, θ_(ω), and θ_(2ω) are the refractive indices and internalpropagation angles at the fundamental and second harmonic wavelengths,respectively.

The nonlinearity profiles of S1 and S2 ared_(S1)(z)=d_(A)(z)−d_(B)(−z+L_(G)) and d_(S2)(z)=d_(A)(z) +d_(B)(z−L),respectively, where L_(G) is the thickness of the space between sample Aand sample B (which can contain an index-matching gel), L=L_(B)+L_(G),and L_(B) is the thickness of sample B. In the expression for d_(S1)(z),since sample B is flipped over, its nonlinearity profile has a negativesign. The physical reason for this sign change is that during poling thesymmetry of the intrinsic material is broken along the z direction.

The MF curves of S1 and S2 are proportional to the square of the FTmagnitude of d_(S1)(z) and dS₂(z), respectively:MF _(S1) =|D _(A)|² +|D _(B)|²−2|D _(A) ∥D _(B)cos(φ_(A)+φ_(B)+φ₁)  (28)MF _(S2) =|D _(A)|² +|D _(B) ^(|2)+2|D _(a) ∥D _(B)cos(φ_(A)−φ_(B)+φ₂)  (29)where φ_(A) and φ_(B) are the Fourier transform phases of d_(A)(z) andd_(B)(z), respectively, φ₁(ƒ)=2πƒL_(G), and φ₂(ƒ)=2πL. All quantitiesdepend on the spatial frequency, but this dependence is omitted forclarity.

The procedure to obtain the profiles d_(A)(z) and d_(B)(z) from themeasured MF curves of S1 and S2 is as follows. The first step is tocompute numerically the inverse Fourier transform of the differenceMFS₂−MF_(S1). It can be shown mathematically that the z≦0 portion ofthis inverse Fourier transform equals C₁(z+L_(G))+C₂(z+L), where C₁(z)and C₂(z) are the convolution functions:C ₁(z)=d _(A)(z)*d _(B)(z)  (30)C ₂(z)=d _(A)(z)*d _(B)(−z)  (31)where the convolution operation is defined as:ƒ(z)*g(z)=∫ƒ(β)·g(z−β)·dβ.  (32)

In the second step, if L_(B)>2W_(B)+W_(A), the functions C₁(z+L_(G)) andC₂(z+L) do not overlap in z, and both C₁(z) and C₂(z) arestraightforward to recover individually. In the third step, the Fouriertransform phases φ_(A)(ƒ) and φ_(B)(ƒ) are retrieved by computing theFourier transforms of C₁(z) and C₂(z), which are equal to|D_(A)|·|D_(B)|·e^(j[φ) ^(A) ^(+φ) ^(B) ^(])and |D_(A)|·|D_(B)|·e^(j[φ)^(A) ^(−φ) ^(B) ^(]), respectively, then adding and subtracting theFourier transform phases of C₁(z) and C₂(z). The phases φ_(A)(ƒ) andφ_(B)(ƒ) are then inserted into Equations 28 and 29, which are solved toobtain the Fourier transform amplitudes |D_(A)| and |D_(B)|. The finalstep is to take the inverse Fourier transform of the recoveredquantities |D_(A)(ƒ)|e^(jφ) ^(A) ^((ƒ)) and |D_(B)(ƒ)|e^(jφ) ^(B) ^((ƒ))to obtain d_(A)(z) and d_(B)(z). Note that any error in the knowledge ofL and L_(G) translates into an error of half this magnitude in thelocation of the corresponding profile in the z direction, but it has noimpact on the shape and magnitude of the recovered profiles.

When the nonlinear samples are thin enough, the L_(B)>2W_(B)+W_(A)condition stated above is not satisfied. In this case, C₁(z) and C₂(z)can still be recovered by using a slightly different procedure thatutilizes all four MF curves (MF_(A), MF_(B) MF_(S1), and MF_(S2)). Thez≦0 portion of the inverse Fourier transform of {−MF_(S1)+MF_(A)+MF_(B)}equals C₁(z+L_(G)), and the z<0 portion of the inverse Fourier transformof {MF_(S) ₂−MF_(A)−MF_(B)}equals C₂(z+L). This property is used toretrieve the convolution functions C₁(z) and C₂(z), and the rest of theprocedure is the same as described above.

In an exemplary embodiment, two Infrasil wafers (Samples A and B, each25 mm ×25 mm ×1 mm) were thermally poled under nominally identicalconditions (5 kV at approximately 270° C. for 15 minutes). After poling,Sample B was polished down on its cathode side to a thickness L_(B) ofapproximately 100 microns to reduce the spacing between the twononlinear regions in the second composite sample S2, thereby reducingthe frequency of oscillations at high angles in MF_(S2), which wouldmake its measurement unnecessarily difficult. For the MF measurements ofSample A, Sample B, the first composite sample, and the second compositesample, a pair of Infrasil half-cylinders were clamped on each side ofthe wafer to avoid total internal reflection and achieve high incidenceangles.

FIGS. 18A–18D illustrate the measured MF curves shown as open circleswith the second harmonic generation efficiency plotted against theincidence angle squared (θ²), to better illustrate details at highangles. The insets of FIGS. 18A–18D schematically illustrate thegeometry of the various samples. As illustrated by FIG. 18D, MF_(S2)oscillates prominently, as expected, since the two nonlinear regions inS2 are spaced a sizable distance (L approximately equal to 140 microns).As illustrated by FIGS. 18A and 18B, the nonlinearity strength ofpolished Sample B is comparable to that of unpolished Sample A. Thisobservation suggests that there is no significant induced nonlinearityin the bulk of the material or close to the cathode surface.

Processing the measured MF data from the first composite sample and thesecond composite sample was performed as described above for thinnersamples (L_(B)<W_(A)+2W_(B)). For each curve, the measured data points(typically approximately 300 data points) were interpolated to generatemore data points and to improve the spatial resolution in the recoveredprofiles. With approximately 2¹⁵ data points (corresponding to a profileresolution of approximately 0.1 microns), the data processing using onlythe MF data from the first composite sample and the second compositesample took approximately 10 minutes on a 500-MHz computer, as comparedto approximately 4 hours with other embodiments described above inrelation to FIG. 9.

FIG. 19 illustrates the recovered sample nonlinearity profiles forSample A and Sample B. The nonlinear coefficients peak at about onemicron below the anode surface, with values of approximately 0.9 pm/Vfor d_(A)(z) and approximately 1.0 pm/V for d_(B)(z). The two profilesare similar in shape and magnitude, which is expected since they werepoled under identical conditions. The difference between the locationsof the negative peaks may be due to small variations in polingconditions. Since d_(A)(z) and d_(B)(z) are quite similar, it is alsopossible to retrieve these profiles by assuming that they are identicaland applying the embodiment described above in relation to FIG. 9. Thenonlinearity profile recovered in this fashion is also shown in FIG. 19.As expected, this profile is similar to both d_(A)(z) and d_(B)(z). Thepositive peak of this profile is nearly identical to those of d_(A)(z)and d_(B)(z), and the negative peak of this profile provides aneffective average of those of d_(A)(z) and d_(B)(z). This comparisonestablishes that these two embodiments are consistent, and that theembodiment of FIG. 16 can discriminate between slightly differentprofiles and thus offer a greater accuracy. The inferred profiles ofFIG. 19 are also similar to profiles retrieved from similarly poledsamples, confirming that the nonlinearity changes sign and extendsapproximately 45 microns below the anode surface.

Certain embodiments of this method provide convenient consistencychecks. Since d_(A)(z) and d_(B)(z) are now known, the theoretical MFdata curves can be computed for Sample A, Sample B, the first compositesample, and the second composite sample to confirm that they areidentical or similar to the corresponding measured MF data curves. Suchtheoretical MF data curves are illustrated in FIGS. 18A-18D as solidlines. The agreement between the measured and theoretical MF data curvesfor Sample A and Sample B are quite good, even at high incidence angles,except above approximately 89 degrees, where the data dip sharply. Thiscan be due to the residual index mismatch between the silica samples andthe gel between the silica samples. The agreement between the measuredand theoretical MF data curves for the first composite sample and thesecond composite sample are good up to an incidence angle ofapproximately 60–70 degrees. At higher angles, the measured MF_(S1) andMF_(S2) curves fail to show the expected rapid oscillations present inthe theoretical curves. The reason is that at high angles, the Makerfringes oscillate rapidly and cannot be resolved because of the finitedivergence of the laser beam. Instead, several adjacent Maker fringesare excited and averaged out. This mechanism may cause the lowercontrast in the measured MF fringes as compared to the theoretical MFfringes at higher angles in FIGS. 18A and 18B.

In practice, a measured MF data curve does not provide the low and highfrequency ends of the Fourier transform spectrum. During dataprocessing, the resulting abrupt discontinuities in the Fouriertransform data in these regions introduce artificial oscillations in theinferred profiles. Since d_(A)(z) and d_(B)(z), C₁(z), and C₂(z) arezero outside a finite region and are square-integrable, their Fouriertransforms are entire functions, which implies that in principle thewhole Fourier transform can be reconstructed uniquely from the knowledgeof the Fourier transform in a finite frequency range, as described byMillane, referenced herein. One implementation of this principle is thePapoulis-Gerchberg algorithm, described by P. J. S. G. Ferreira in IEEETransactions on Signal Processing, 1994, Volume 42, page 2596,incorporated in its entirety by reference herein, which can be used toextrapolate the measured data into the unmeasured low and high frequencyend portions of the Fourier spectrum.

Although this embodiment utilizes the measurement of two MF data curves,it is not any more labor-intensive than embodiments which utilize onlyone measurement, since it provides two profiles instead of a single one.Furthermore, after a pair of nonlinear samples have been characterizedin this manner, either one of the two samples can be used as a referencesample with a known profile for measuring the profile of any othernonlinear sample, thereby using only a single new MF measurement (e.g.,MF_(S2)). Data processing for this single-measurement case is slightlydifferent and even simpler. For example, if the sample that is measuredis S2, by selecting a thick enough sample so that L_(B)>W_(A)+W_(B),C₂(z) can be retrieved unambiguously from the z≦0 portion of the inverseFourier transform of MF_(S2). Since the nonlinearity profile d_(A)(z) ofthe reference sample is known, the Fourier transform of C₂(z) (i.e.,|D_(A)|·|D_(B)|·e^(j[φ) ^(A) ^(−φ) ^(B) ^(])) immediately provides both|D_(A)| and φ_(B), and taking the inverse Fourier transform of|D_(B)|·e^(jφ) ^(B) yields the unknown profile d_(B)(z).

Embodiments described herein can also be used to measure other physicalfunctions. FIG. 20 is a flowchart of a method 200 in accordance withembodiments of the present invention. In an operational block 210, asymmetric composite function is formed. In an operational block 220,aFourier transform of the symmetric composite function is obtained. In anoperational block 230,an inverse Fourier transform of the obtainedFourier transform is calculated. The calculated inverse Fouriertransform provides information regarding the physical function.

In certain embodiments, the symmetric composite function is an oddfunction (i.e., is symmetric about the origin). In other embodiments,the symmetric composite function is even (i.e., symmetric about they-axis). In certain embodiments, obtaining the Fourier transform of thecomposite function comprises obtaining a Fourier transform magnitude ofthe composite function and using the Fourier transform magnitude tocalculate the Fourier transform of the composite function.

For example, instead of forming a symmetric composite function in thespatial domain as described above in relation to the nonlinearityprofile of poled silica, other embodiments form a symmetric intensityprofile in the time domain by the use of time reversal. In suchembodiments, the symmetric (even) composite function can have utilitywhere phase information is needed but unavailable (e.g., ultra-shortpulse diagnosis using auto-correlators). An example of time reversal isdescribed by D. A. B. Miller in Time Reversal of Optical Pulses byFour-Wave Mixing, Optics Letters Vol. 5, 1980, pages 300–302, which isincorporated in its entirety by reference herein.

FIG. 21 is a flow diagram of a method 400 of determining the temporalwaveform 312 of a laser pulse 310 in accordance with embodiments of thepresent invention. FIG. 22 schematically illustrates one configurationfor utilizing four-wave mixing (FWM) to provide the time-reversed pulse320 in accordance with embodiments of the present invention. Otherconfigurations are also compatible with the method of FIG. 21. Referringto the elements illustrated in FIG. 22, in an operational block 410, alaser pulse 310 is provided. The laser pulse 310 has a temporal waveform312. In an operational block 420, a time-reversed pulse 320 is provided.The time-reversed pulse 320 has a time-reversed temporal waveform 322which corresponds to the temporal waveform 312 after beingtime-reversed. In an operational block 430, the temporal waveform 312 ofthe laser pulse 310 and the time-reversed temporal waveform 322 of thetime-reversed pulse 320 form a symmetric composite waveform. In anoperational block 440, a Fourier transform of the symmetric compositewaveform is obtained. In certain embodiments, obtaining the Fouriertransform of the symmetric composite function comprises measuring aFourier transform magnitude of the symmetric composite function, andusing the measured Fourier transform magnitude to calculate the Fouriertransform of the symmetric composite function. In an operational block450, an inverse Fourier transform of the obtained Fourier transform iscalculated. The calculated inverse Fourier transform providesinformation regarding the temporal waveform 312 of the laser pulse 310.

FWM has been a subject of interest in applications such as aberrationcompensation, spatial information processing, frequency filtering, pulseenvelope shaping, and dispersion compensation. As illustrated in FIG.22, a nonlinear medium 330 of length L is pumped by two pulsed pumpwaves 340, 350. An input laser pulse 310 (with temporal waveform 312given by E_(p)(x,t)) launched into the nonlinear medium 330 generates aphase conjugate pulse 320 (with time-reversed temporal waveform 322given by E_(c)(x,t)), which is the time-reversed version of the temporalwaveform 312 of the input pulse 310. In the embodiment described below,the input pulse 310, two pump waves 340, 350, and the nonlinear mediumare at the same place for the duration of the input pulse 310. Inaddition, the input pulse 310, and the two pump waves 340, 350 overlapin the frequency domain.

Illustratively, the temporal waveform 312 of the input pulse 310 can bewritten in the following form:

$\begin{matrix}{{E_{p}\left( {x,t} \right)} = {{\frac{1}{2}{u_{p}(t)}{\mathbb{e}}^{j\;{({{\omega_{p}t} - {kx}})}}} + {{complex}\mspace{14mu}{{conjugate}.}}}} & (33)\end{matrix}$where u_(p)(t) is the modulation of the carrier e^(j(ω) ^(p) ^(t-kx)).The Fourier transform of u_(p)(t) has the following form:Ū _(p)(ω)=∫u _(p)(t)e^(−jωt) dt.  (34)The temporal waveform of the resultant conjugate pulse 320 has thefollowing form:

$\begin{matrix}{{E_{c}\left( {x,t} \right)} = {{\frac{1}{2}{u_{c}(t)}{\mathbb{e}}^{j{({{\omega_{c}t} + {kx}})}}} + {{complex}\mspace{14mu}{conjugate}}}} & (35)\end{matrix}$where “c” stands for “conjugate.” Note that the k-vector of theconjugate pulse E_(c)(x,t) has the reverse sign as expected. The Fouriertransform of the envelope function u_(c)(t) is defined the same way:Ū _(c)(ω)=∫u _(c)(t)e^(−jωt) dt.  (36)The relationship between the carrier frequencies ω_(c), ω_(p), asdefined above, and the center frequencies ∫_(pump,1) and ω_(pump,2) ofthe two pumps 340, 350 is:ω_(pump,1)+ω_(pump,2)−ω_(p)=ω_(c)  (37)With these definitions, the envelope function u_(c)(t) can be expressedas:u _(c)(t)=∫h(ω)Ū*_(p)(−ω)e^(jωt) dω  (38)where h(ω) is the response function of the nonlinear material 330. Forbroadband conjugators (with respect to the spectrum of u_(p)(t)), h(ω)can be taken as a constant (K), giving u_(c)(t) the following form:u _(c)(t)=K ∫Ū* _(p)(−ω)e^(jωt) dω.  (39)The foregoing forms of the envelope function u_(c)(t) were obtainedusing the teachings of R. A. Fischer et al. in Transient Analysis ofKerr-Like Phase Conjugators Using Frequency-Domain Techniques, PhysicalReview A, Vol. 23, 1981, pages 3071–3083, which is incorporated in itsentirety by reference herein.

The above equations can be used to show that for continuous-wave (CW)pumping, FWM can not time-reverse pulsed fields. This property of FWM inCW pumping has been extensively studied for dispersion compensation infiber links. Examples of such work include A. Yariv et al. inCompensation for Channel Dispersion by Nonlinear Optical PhaseConjugation, Optics Letters Vol. 4, 1979, pages 52–54, and S. Watanabeet al. in Compensation of Chromatic Dispersion in a Single-Mode Fiber byOptical Phase Conjugation, IEEE Photonics Technical Letters, Vol. 5,1993, pages 92–95, both of which are incorporated in their entirety byreference herein.

However, pulsed pumping can be used to achieve time reversal ofamplitude pulses. Following the derivation of D.A.B. Miller (citedabove), u_(c)(t) can be expressed as:u _(c)(t)=K′u* _(p)(−t+τ ₀)  (40)where K′ and τ₀ are constants. The −t term in Equation 40 indicates thetime reversal operation. Note that Equation 33 and Equation 37 are stillvalid for this case. The assumptions made in the derivation of Equation40 are that:

-   -   (1) the nonlinear medium 330 has a length L equal to or longer        than the spatial length of the input pulse 310 (i.e., large L        assumption) so that the input pulse 310 is completely within the        nonlinear medium 330 at some time during the interaction;    -   (2) the pump pulses 340, 350 are perpendicular to the nonlinear        medium as shown in FIG. 22;    -   (3) pump pulses 340, 350 are short compared with the input pulse        310 (i.e., the spectra of both pump pulses 340, 350 are broad        enough so that all of the frequency components of the input        pulse 310 see a substantially uniform pump spectral power        density for both pumps);    -   (4) as a consequence of (3), the thickness of the nonlinear        medium 330 is preferably equal to or slightly greater than the        spatial length of the pump pulses 340, 350); and    -   (5) the timing of the pulses is such that when pulse 310        spatially fills the nonlinear medium 330 (i.e. the input pulse        310 is fully within the medium), the pump pulses 340 and 350 are        timed to be overlapping in space with the input pulse 310 across        the nonlinear medium filled by the input pulse.        Some of the details of these assumptions can be found        in D. A. B. Miller's work (cited above). As described below, the        apparatus schematically illustrated by FIG. 22 can serve as a        phase conjugate mirror which generates the time-reversed        waveform corresponding to an input waveform.

In other embodiments, the time-reversed pulse 320 can be provided usingholographic techniques in accordance with embodiments of the presentinvention. Femtosecond spectral holography can be considered as atemporal analog of the classical spatial domain holography. In classicalholography, a spatial information carrying beam (signal) and a uniformreference beam interfere in a recording medium, thereby recording ahologram comprising a set of fringes. Illumination of the hologram witha uniform test beam reconstructs either a real or conjugate image of thesignal beam, depending on the geometry.

Femtosecond spectral holography for the time reversal process comprisesa recording phase and a read-out phase, as schematically illustratedrespectively by FIGS. 23A and 23B. In the recording phase (FIG. 23A),the reference pulse 360 is short with a relatively flat and broadspectrum, and the input pulse 310 has a temporal waveform 312 that has ashape indicative of information carried by the input pulse 310. Duringthe recording of the hologram, a grating 370 is used to disperse boththe reference pulse 360 and the input pulse 310 into their spectralcomponents, which then propagate through a collimating lens system 372.The interference pattern formed by the complex amplitudes of eachspectral component is then recorded in the holographic recording plate375.

In the read-out phase (FIG. 23B), a short test pulse 380 is dispersed bythe grating 370 and then recollimated by the lens 372. The illuminationof the holographic plate 375 with this recollimated dispersed test beam392 produces the beam 396. Using the lens 387 for recollimation and thesecond grating 385, a time-reversed replica 390 of the original inputpulse 310 is produced. Also as a by-product, the transmitted test beam392 appears at the output. The details of this technique are describedmore fully by A. M. Weiner et al. in Femtosecond Spectral Holography,IEEE Journal of Quantum Electronics, Vol. 28, 1992, pages 2251–2261, andA. M. Weiner et al. in Femtosecond Pulse Shaping for Synthesis,Processing and Time-to-Space Conversion of Ultrafast Optical Waveforms,IEEE Journal of Selected Topics in Quantum Electronics, Vol. 4, 1998,pages 317–331, both of which are incorporated in their entireties byreference herein.

The envelope of the output pulse 390 can be expressed as:u _(out)(t)≈u _(t)(−t)*u _(r)(−t)*u _(s)(t)e ^(j{overscore (K)}) ^(t)^({overscore (r)}) +u _(t)(t)*u _(r)(t)*u _(s)(−t)e ^(j{overscore (K)})² ^({overscore (r)})  (41)where u_(out)(t), u_(t)(t), u_(r)(t), and u_(s)(t) are the complexenvelope functions of the electric fields of the output pulse 390, testpulse 380, reference pulse 360, and input pulse 310, respectively. Thesign ‘*’ denotes the convolution function, and {overscore(K)}₁={overscore (k)}_(t)−{overscore (k)}_(r)+{overscore (k)}_(s) and{overscore (K)}₂={overscore (k)}_(t)+{overscore (k)}_(r)−{overscore(k)}_(s).

When the test pulse 380 and the reference pulse 360 are considerablyshorter than the input pulse 310, the complex envelope functionsu_(s)(t), u_(t)(t), and u_(r)(t) will act as delta functions withrespect to u_(s)(t), modifying the envelope of the output pulse 390 tobe:u _(out)(t)≈u _(s)(t)e ^(j{overscore (K)}) ¹ ^({overscore (r)}) +u_(s)(−t) e ^(j{overscore (K)}) ² ^({overscore (r)})  (42)Therefore, as a result of the illumination of the holographic plate withthe test pulse 380, the output pulse 390 serves as the time-reversedsignal pulse 320 in the {overscore (K)}₂ direction. As described below,the apparatus schematically illustrated by FIGS. 23A and 23B can serveas a phase conjugate mirror which generates the time-reversed waveformcorresponding to an input waveform.

Note that embodiments of both the pulse-pumped FWM and the spectralholography techniques use shorter pulses than the input pulse 310 totime-reverse the input pulse 310. For pulse-pumped FWM, shorter pumppulses 340, 350 are used, and for holography, shorter test pulses 380and reference pulses 360 are used.

FIG. 24 schematically illustrates one embodiment of a classicalintensity correlator utilizing a Michelson interferometer 500 incombination with a second-harmonic generating (SHG) crystal 510 toestimate the temporal waveform of a laser pulse. The input pulse 310 issplit into a first pulse 520 and a second pulse 522 by the beamsplitter530. The first pulse 520 is reflected by a fixed mirror 550 back throughthe beamsplitter 530, thereby transmitting a third pulse 524. The secondpulse 522 is reflected by a movable mirror 552 back through thebeamsplitter 530, thereby transmitting a fourth pulse 526 towards thefocusing lens 540. The movable mirror 552 adds a variable delay τ to thefourth pulse 526 with respect to the third pulse 524. The focusing lens540 focuses the third pulse 524 and the fourth pulse 526 onto the SHGcrystal 510. The SHG crystal 510 generates a second-harmonic (SH) field538 which is a cross-correlation produ_(c)t of the third pulse 524 andthe fourth pulse 526. Other SH fields 534, 536 generated by the SHGcrystal 510 do not carry information regarding the third and fourthpulses 524, 526 simultaneously, and these other SH fields are blocked bythe pinhole 560 placed after the SHG crystal 510. After passing througha filter 570 which blocks light at the fundamental wavelength and whichpasses the SH light, the cross-correlation product 538 is detected by aphotomultiplier tube 580.

The third pulse 524 emerging from one arm of the Michelsoninterferometer 500 has a waveform given by:E ₁(t)=u₁(t)e ^(jωt)  (43)where u₁(t) is the complex envelope function of the input pulse 310. Thefourth pulse 526 emerging from the other arm of the Michelsoninterferometer 500 has a waveform given by:E ₂(t)=u ₂(t)e ^(jωt) =u ₁(t−τ)e ^(jω(t−τ))  (44)where τ is the relative time delay between the third pulse 524 and thefourth pulse 526 imposed by the delay τ between the two arms. Because ofthe focusing lens 540, the third pulse 524 and the fourth pulse 526 havedifferent k-vectors at the surface of the SHG crystal 510. Consequently,the SH field 538 generated by the SHG crystal 510 that is not blocked bythe pinhole 560 includes only terms due to the interaction of the thirdpulse 524 and the fourth pulse 526, and has a waveform given by:E _(2ω)(t)=u _(2ω)(t)e ^(j2ωt) =ηu ₁(t)u ₂(t)e ^(j2ωt)=ηu_(t)(t)u₁(t−τ)e^(−jωτ) e ^(j2ωt)  (45)where τ is a conversion efficiency factor corresponding to, the SHGcrystal and the system geometry. The signal detected by thephotomultiplier 580 (i.e., the autocorrelation function) is given by:A _(PMT)(τ)=∫E _(2ω)(t)E* _(2ω)(t)dt=∫|η| ² |u ₁(t)|² |u ₁(t−τ)|²dt=|η|² ∫I(t) I(t−τ)dt  (46)where I(t)=|u₁(t)|² is the input signal intensity. In general, η is afunction of frequency and the functions u₁(t) and u₂(t) can be expandedas plane waves with Fourier transform amplitudes (i.e.,u_(i)(t)=∫Ū_(i)(ω)e^(jωt)dω). However, it is assumed here that η isconstant over the frequency range of the combined pulses, which isequivalent to assuming that χ⁽²⁾ is independent of frequency. Thisassumption holds when the effective crystal length is shorter than thecoherence length of the harmonic generation over the pulse bandwidth.

By taking the Fourier transform of both sides of Equation 46 (the signaldetected by the photomultiplier 580 for a single pulse) provides theFourier transform of the autocorrelation function and is given by:Ā _(PMT) _(—) _(single)(ƒ)=|τ|² |Ī(ƒ)| ²  (47)where Ā_(PMT) _(—) _(single)(ƒ) and Ī(ƒ) are the Fourier transforms ofA_(PMT)(τ) and I(τ), respectively. Thus, the signal from the Michelsoninterferometer 500 of FIG. 24 provides the magnitude of the Fouriertransform of the input pulse intensity I(t).

However, the magnitude of the Fourier transform is not sufficientinformation to retrieve uniquely the input pulse intensity I(t). To doso would also require the knowledge of the phase of the Fouriertransform, which this classical autocorrelation method does not provide.This difficulty in uniquely determining the input pulse intensity I(t)is analogous to that of retrieving the second-order susceptibilityspatial profile of a nonlinear material, as described above. Inpractice, a number of methods can be used to avoid this difficulty. Forexample, a certain shape for the pulse intensity (e.g., a Gaussian) canbe assumed, but there is no way to independently verify that the assumedshape is the true temporal waveform. This method therefore provides onlyan estimate of the pulse shape.

By using a time-reversal technique compatible with embodiments of thepresent invention (e.g., pulse-pumped FWM or femtosecond spectralholography), a time-reversed pulse with a time-reversed temporalwaveform I(−t) of the temporal waveform I(t) of an arbitrary pulse canbe produced. A symmetric composite waveform can then be formed bydelaying the time-reversed pulse with respect to the original pulse andcombining the temporal waveform with the time-reversed temporalwaveform, for example with a beamsplitter.

FIG. 25 schematically illustrates a general configuration 600 forconverting a periodically repeated sequence of pulses into aperiodically repeated sequence of symmetric pulses using a movable phaseconjugative mirror 610. The input pulse sequence 620 comprises a seriesof original pulses 622. The original pulses 622 can be symmetric orasymmetric. Each of these original pulses 622 is split by thebeamsplitter 630 into two portions 640, 642, with one portion 640 beingreflected by a phase conjugate mirror 610. In certain embodiments, thephase conjugate mirror 610 comprises an apparatus as schematicallyillustrated by FIG. 22 in which the input pulse 310 is used to generatethe output pulse 320. In certain other embodiments, the phase conjugatemirror 610 comprises an apparatus as schematically illustrated by FIGS.23A and 23B in which the input pulse 310 is used to generate the outputpulse 390. The portions 640, 642 are combined to form a symmetriccomposite waveform 650 comprising the temporal waveforms of the originalpulses 622 and of the corresponding phase conjugate (time-reversed)pulses 652. The amount of delay At between an original pulse 622 and thecorresponding time-reversed pulse 652 in the symmetric pulse sequence650 is adjustable by moving the phase conjugative mirror 610.

The symmetric composite waveform 650 has the following form:

$\begin{matrix}{{I_{symmetric}(t)} = {{I\left( {{- t} - \frac{\Delta\; t}{2}} \right)} + {I\left( {t - \frac{\Delta\; t}{2}} \right)}}} & (48)\end{matrix}$where Δt is the variable time delay between the time-reversed pulse 652and the original pulse 622. The Fourier transform of Equation 48 isgiven by:

$\begin{matrix}{{{\overset{\_}{I}}_{symmetric}(f)} = {{{\overset{\_}{I}\left( {- f} \right)}{\mathbb{e}}^{j\; f\frac{\Delta\; t}{2}}} + {{\overset{\_}{I}(f)}{{\mathbb{e}}^{{- j}\; f\frac{\Delta\; t}{2}}.}}}} & (49)\end{matrix}$Since I(t) is real, |Ī(ƒ)|=|Ī(−ƒ)|and φ(ƒ)=−φ(ƒ), where Ī(ƒ) has beendefined as: Ī(ƒ)=|Ī(ƒ)|e^(jφ(ƒ)). Using these identities, together withEquations 47 and 49, the Fourier transform of the autocorrelationfunction corresponding to the symmetric composite waveform 650 has thefollowing form:Ā _(PMT) _(—) _(double)(ƒ)=2|τ|² Ī(ƒ)| ²|1+cos (2φ(ƒ)−ƒΔt)|.  (50)

In embodiments in which Ā_(PMT) _(—) _(double) is real and greater thanzero for all frequencies ω (i.e., there are no zero crossings of Ā_(PMT)_(—) _(double) from the cosine term), then the inverse Fourier transformof Equation 50 provides the intensity of the symmetric pulse sequence650 without any ambiguity. Once the intensity of the symmetric pulsesequence 650 is calculated in this way, the intensity of the originalpulse can be found by separating the pulse 650 in the middle.

FIG. 26 illustrates an exemplary temporal waveform of an input pulse 622compatible with embodiments of the present invention. Theautocorrelation technique commonly used to estimate the temporal profileof an optical pulse is applied to this input pulse as follows. By usingthe configuration schematically illustrated in FIG. 24, theautocorrelation function (Equation 46) corresponding to the temporalwaveform is measured. The magnitude of the Fourier transform of thesignal Ā_(PMT) _(—) _(single) τ recorded by the photomultiplier tube 580is schematically illustrated in FIG. 27. The frequency ƒ shown in FIG.27 is 1/τ, and is not to be confused with the optical frequency.

In the following, an embodiment is illustrated analytically by showingthe result of numerical simulations using the input pulse of FIG. 26.First, the time-reversal of the input pulse is obtained using one of thephase conjugate mirror schemes described above. FIG. 28 illustrates thesymmetric composite waveform 650 defined by Equation 48, which isobtained by combining the input pulse and the time-reversed pulse. Thesymmetric composite waveform 650 comprises the temporal waveform of theoriginal pulse 622 of FIG. 25 plus the time-reversed temporal waveformof the corresponding time-reversed pulse 652, separated by some timedelay.

In the second step, the autocorrelator of FIG. 24 is used to generatethe autocorrelation function corresponding to the symmetric compositewaveform 650. FIG. 29 illustrates the magnitude of Ā_(PMT) _(—)_(double (ƒ))(expressed by Equation 50), which is the Fourier transformof the measured Ā_(PMT) _(—) _(double(τ)). The envelope of the Fouriertransform magnitude of the measured Ā_(PMT) _(—) _(double(τ)) is ascaled version of the Fourier transform magnitude (as shown in FIG. 27)of the signal Ā_(PMT) _(—) _(double(τ))measured by the photomultipliertube 580. Therefore, |Ī(ƒ)|² or |Ī(ƒ)| can be recovered from only theenvelope of the Fourier transform magnitude of the measured Ā_(PMT) _(—)_(double(τ))(i.e., from only the measurement of the autocorrelationfunction of the symmetric composite waveform 650). This result impliesthat the measurement of the autocorrelation function for the singlepulse as schematically illustrated in FIG. 26 is redundant. However,since it does not add complexity to the measurement, certain embodimentsalso comprise obtaining |Ī(ƒ)|² data from a separate source for errorchecking purposes.

In the third step, the function g(ƒ)=1+cos (2φ(ƒ)−ƒΔt) is determined bydividing both sides of Equation 50 by |Ī(ƒ)|². From the knowledge ofg(ƒ)−1, the function

${\cos\left( {{\varphi(f)} - {f\;\frac{\Delta\; t}{2}}} \right)}$can be determined. Using Equation 49, the Fourier transform of thesymmetric temporal waveform can be expressed as:

$\begin{matrix}{{{\overset{\_}{I}}_{symmetric}(f)} = {2{{\overset{\_}{I}(f)}}\;{{\cos\left( {{\varphi(f)} - {f\;\frac{\Delta\; t}{2}}} \right)}.}}} & (51)\end{matrix}$Therefore, since |Ī(ƒ)| is known from FIG. 25 or from the envelope ofFIG. 29, the only information needed to determine I_(symmetric)(t)directly by taking the inverse Fourier transform of Equation 51 is theknowledge of

${\cos\left( {{\varphi(f)} - {f\;\frac{\Delta\; t}{2}}} \right)}.$

To determine the function

$\cos\left( {{\varphi(f)} - {f\;\frac{\Delta\; t}{2}}} \right)$from the function

${{\cos\left( {{\varphi(f)} - {f\;\frac{\Delta\; t}{2}}} \right)}},$two possible cases can be analyzed. In the first case, if there are nozero crossings of the term

${{\cos\left( {{\varphi(f)} - {f\;\frac{\Delta\; t}{2}}} \right)}},$then there is no ambiguity due to the absolute value sign sinceintensity has to be non-negative (−I_(symmetric)(t) is not a possiblesolution). In the second case, if there are some zero crossings of theterm

${{\cos\left( {{\varphi(f)} - {f\;\frac{\Delta\; t}{2}}} \right)}},$the sign ambiguities of the cosine term between the zero crossings canbe removed by using a property of Fourier transforms. For a real andsymmetric function such as I_(symmetric)(t), the Fourier transformĪ_(symmetric)(ƒ) is equivalent to the Hartley transform I_(symmetric)^(Ha)(ƒ). Therefore, the magnitude of the Hartley transform ofI_(symmetric)(t) (i.e.,

$\left( {{i.e.},{{{I_{symmetric}^{Ha}(f)}} = {{{{\overset{\_}{I}}_{symmetric}(f)}} = {2{{\overset{\_}{I}(f)}}{{\cos\left( {{\varphi(f)} - {f\;\frac{\Delta\; t}{2}}} \right)}}}}}} \right)$can be determined from the knowledge of

${{\cos\left( {{\varphi(f)} - {f\;\frac{\Delta\; t}{2}}} \right)}}\mspace{14mu}{and}\mspace{14mu}{{{\overset{\_}{I}(f)}}.}$For a real and compact support function (i.e., one that equals zerooutside a finite region), such as I_(symmetric)(t), the intensity of theHartley transform is enough to uniquely recover the original function.See, e.g., N. Nakajima in Reconstruction of a real function from itsHartley-transform intensity, Journal of the Optical Society of AmericaA, Vol. 5, 1988, pages 858–863, and R. P. Millane in Analytic Propertiesof the Hartley Transform and their Implications, Proceedings of theIEEE, Vol. 82, 1994, pages 413–428, both of which are incorporated intheir entirety by reference herein.

FIG. 30A illustrates the recovered symmetric temporal waveform and FIG.30B illustrates the difference between the recovered symmetric temporalwaveform and the temporal waveform of the original pulse (as shown inFIG. 28). As illustrated by FIG. 30B, the two waveforms are in excellentagreement (within approximately 0.15) with one another. The differenceshown in FIG. 30B is in fact just a numerical calculation artifact,which can be improved with increased accuracy. Thus, the configurationschematically illustrated in FIG. 24 is capable of recoveringultra-short temporal waveforms unambiguously when used with atime-reversal scheme, su_(c)h as that illustrated by FIG. 25.

FIG. 31 schematically illustrates a system 700 for another embodimentfor determining the temporal waveform of a laser pulse. The input pulse710 impinges onto a grating 720 which disperses the input pulse 710 intoits spectral components. A lens 730 recollimates the spectral componentsand images them onto different elements of a CCD imaging device 740.

An arbitrary input pulse 710 has the following form:u _(s)(t)={tilde under (u)}_(s)(t)e ^(jω) ^(c) ^(t)  (52)where {tilde under (u)}_(s)(t) is the complex envelope function andω_(c) is the carrier frequency. Equation 52 can be rewritten as:u _(s)(t)=∫{tilde under (U)} _(s)(ω−ω_(c))e ^(jωt) dω  (53)where {tilde under (U)}_(s)(ω) denotes the Fourier transform of i,(t).The input pulse 710 is decomposed by the grating 720 into severalmonochromatic plane waves with amplitudes {tilde under(U)}_(s)(ω−ω_(c)). By finding the response of the system 700 to eachindividual harmonic component (i.e. {tilde under(U)}_(s)(ω−ω_(c))e^(jωt)), the overall response of the system 700 can bedetermined using the integral given in Equation 53.

The field of a single harmonic {tilde under (U)}_(s)(ω−ω_(c))e^(jωt) atthe plane 722 immediately after being dispersed by the grating 720 canbe written in the following form:

$\begin{matrix}{{{\overset{\sim}{U}}_{1s}\left( {{x;\omega},t} \right)} = {{{\overset{\sim}{U}}_{s}\left( {\omega - \omega_{c}} \right)}{\mathbb{e}}^{j\;\omega\; t}{w(x)}{\mathbb{e}}^{{- j}\;{x{({\frac{\omega - \omega_{c}}{c}\sin\mspace{11mu}\theta})}}}}} & (54)\end{matrix}$where w(x) is the pupil function of the optical field on the grating720, c is the speed of light, x is the coordinate along the plane 722,and θ is the incident angle of the input pulse 710 to the grating 720.This form of the single harmonic field is described by P. C. Sun et al.in Femtosecond Pulse Imaging: Ultrafast Optical Oscilloscope, Journal ofthe Optical Society of America, Vol. 14, 1997, pages 1159–1170, which isincorporated in its entirety by reference herein.

The last exponential term of Equation 54 accounts for the diffractionexperienced by the spectral components of the input pulse 710 due to thegrating 720, assuming only first-order diffraction. The grating 720 isarranged such that the first diffraction order for the spectralcomponent at ω=ω_(c) propagates along the direction of the optical axisof the system 700. The lens 730 transforms the image at the plane 722into an image at plane 742. The fields at the two planes 722, 742 arerelated by a spatial Fourier transform. By taking the spatial Fouriertransform of Equation 30, the field at the plane 742 can be written as:

$\begin{matrix}{{{\overset{\sim}{U}}_{2s}\left( {{f_{x^{\prime}};\omega},t} \right)} = {{{\overset{\sim}{U}}_{s}\left( {\omega - \omega_{c}} \right)}{\mathbb{e}}^{j\;\omega\; t}{W\left( {f_{x^{\prime}} + {\frac{\omega - \omega_{c}}{2\;\pi\; c}\sin\mspace{11mu}\theta}} \right)}}} & (55)\end{matrix}$where W(ƒ_(x)′) is the spatial Fourier transform of w(x). x′ is thecoordinate along the plane 742, and ƒ_(x′) is the spatial frequencywhich can be written as:

$\begin{matrix}{f_{x^{\prime}} = \frac{\omega\; x^{\prime}}{2\;\pi\;{cF}}} & (56)\end{matrix}$where F is the focal length of the lens.

The total response of the system 700 is the spectral decomposition ofthe field u_(s)(t) of the input pulse 710 and can be found byintegrating Equation 55 over the frequency range, i.e.,:u _(2s)(x′;t)=∫{tilde under (U)} _(2s)(x′;ω,t)dω  (57)By using the paraxial approximation and by assuming a large illuminationwindow w(x), the total response given by Equation 57 can be simplifiedto the following form:

$\begin{matrix}{{u_{2s}\left( {x^{\prime};t} \right)} \approx {{{\overset{\sim}{U}}_{s}\left( {- \frac{x^{\prime}\omega_{c}}{F\mspace{11mu}\sin\;\theta}} \right)}{w\left( {\left\lbrack {1 - \frac{x^{\prime}}{F\mspace{11mu}\sin\;\theta}} \right\rbrack\frac{ct}{\sin\mspace{11mu}\theta}} \right)}{{\mathbb{e}}^{j\;{\omega_{c}{({{\lbrack{1 - \frac{x^{\prime}}{F\mspace{11mu}\sin\;\theta}}\rbrack}t})}}}.}}} & (58)\end{matrix}$

By generating a phase conjugate pulse as schematically illustrated inFIG. 25, the total output waveform can be expressed as:u_(total)(t)=u_(s)(t)+u_(PC)(t−Δt), where PC denotes “phase conjugate.”The function u_(pc)(t) is dependent on the type of phase conjugatemirror used to generate the phase conjugate pulses. For phase conjugatepulses formed using pulse-pumped FWM, the total field u_(total)(t) canbe expressed, using Equation 41, in the following form:u _(total)(t)=({tilde under (u)} _(s)(t)+{tilde under (u)} _(s)*(−t−Δt))e ^(jω) ^(c) ^(t)  (59)where Δt is the time delay between the original pulse and itscorresponding phase conjugate pulse. Feeding this total field into thesystem 700 illustrated in FIG. 31, the resultant image field at theplane 742 of the CCD imaging device 740, using Equation 58, can beexpressed as:

$\begin{matrix}{{u_{{total},2}\left( {x^{\prime};t} \right)} \approx {{{{\overset{\sim}{U}}_{s}\left( {- \frac{x^{\prime}\omega_{c}}{F\mspace{11mu}\sin\;\theta}} \right)}{w\left( {\left\lbrack {1 - \frac{x^{\prime}}{F\mspace{11mu}\sin\;\theta}} \right\rbrack\frac{ct}{\sin\mspace{11mu}\theta}} \right)}{\mathbb{e}}^{j\;{\omega_{c}{({{\lbrack{1 - \frac{x^{\prime}}{F\mspace{11mu}\sin\;\theta}}\rbrack}t})}}}} + {{{\overset{\sim}{U}}_{s}^{*}\left( {- \frac{x^{\prime}\omega_{c}}{F\mspace{11mu}\sin\;\theta}} \right)}{w\left( {\left\lbrack {1 - \frac{x^{\prime}}{F\mspace{11mu}\sin\;\theta}} \right\rbrack\frac{ct}{\sin\mspace{11mu}\theta}} \right)}\;{\mathbb{e}}^{j\;{\omega_{c}{({{\lbrack{1 - \frac{x^{\prime}}{F\mspace{11mu}\sin\;\theta}}\rbrack}t})}}}{{\mathbb{e}}^{j\;\Delta\;{t{({- \frac{x^{\prime}\omega_{c}}{F\mspace{11mu}\sin\;\theta}})}}}.}}}} & (60)\end{matrix}$

The CCD imaging device 740 at the plane 742 is responsive to intensity,which can be written as:I(x′)=∫|u_(total,2)(x′;t)|² dt  (61)Defining {tilde under (U)}_(s)(ω)=|{tilde under (U)}_(s)(ω)|e^(jφ(ω))and A=−F sin θ, Equation 61 can be rewritten as:

$\begin{matrix}{{I_{double}\left( x^{\prime} \right)} = {2\left( {\int{{{w\left( {\left\lbrack {1 + \frac{x^{\prime}}{A}} \right\rbrack\frac{ct}{\sin\mspace{11mu}\theta}} \right)}}^{2}{\mathbb{d}t}}} \right){{{\overset{\sim}{U}}_{s}\left( \frac{x^{\prime}\;\omega_{c}}{A} \right)}}^{2}{{{1 + {\cos\left( {{2\;{\Phi\left( \frac{x^{\prime}\omega_{c}}{A} \right)}} - {\Delta\; t\frac{x^{\prime}\omega_{c}}{A}}} \right)}}}.}}} & (62)\end{matrix}$Furthermore, Equation 62 can be rewritten as:

$\begin{matrix}{{I_{double}\left( x^{\prime} \right)} = {2{G\left( x^{\prime} \right)}{{{\overset{\sim}{U}}_{s}\left( \frac{x^{\prime}\;\omega_{c}}{A} \right)}}^{2}{{1 + {\cos\left( {{2\;{\Phi\left( \frac{x^{\prime}\omega_{c}}{A} \right)}} - {\Delta\; t\frac{x^{\prime}\omega_{c}}{A}}} \right)}}}}} & (63)\end{matrix}$with

${G\left( x^{\prime} \right)} = {\int{{{w\left( {\left\lbrack {1 + \frac{x^{\prime}}{A}} \right\rbrack\frac{ct}{\sin\mspace{11mu}\theta}} \right)}}^{2}{{\mathbb{d}t}.}}}$Equation 63 is very similar to Equation 50, which was obtained forembodiments utilizing the intensity correlator of FIG. 24. Furthermore,the intensity profile on the CCD imaging device 740 for a single pulseu_(s)(t) can be expressed as:

$\begin{matrix}{{I_{single}\left( x^{\prime} \right)} = {{G\left( x^{\prime} \right)}{{{{\overset{\sim}{U}}_{s}\left( \frac{x^{\prime}\omega_{c}}{A} \right)}}^{2}.}}} & (64)\end{matrix}$Equation 64 is also very similar to Equation 47, which was derived abovein relation to the intensity correlator embodiment as shown in FIG. 24.

To recover the complex envelope function {tilde under (u)}_(s)(t), asdefined in Equation 52, the same algorithm described above forrecovering I(t) can be applied. This process is illustrated in the FIGS.32 through 37 using computer-generated simulations. FIGS. 32A and 32Billustrate the magnitude (intensity) and phase of an arbitraryasymmetric complex envelope function {tilde under (u)}_(s)(t) to becharacterized, respectively. (FIG. 32A is the same input pulse as shownin FIG. 26). FIG. 33 illustrates the intensity profile for this complexfield, including the carrier frequency oscillations. Note that FIG. 26described above did not include the carrier frequency oscillations andwas just the envelope of the intensity. In a first step, the generalconfiguration 600 of FIG. 25 is used to generate the symmetric temporalwaveform whose complex envelope function is defined by Equation 59. FIG.34 illustrates the symmetric temporal waveform, including the carrierfrequency, generated by this first step. The symmetric temporal waveformcomprises the temporal waveform of the original pulse plus thetime-delayed temporal waveform of the time-reversed pulse.

In a second step, the system of FIG. 31 is used to measure the Fouriertransform magnitude of the symmetric pulse with the CCD imaging device.FIG. 35A illustrates the detected intensity I(x′) on the CCD imagingdevice at the plane 742 for the symmetric pulse of FIG. 34. Theintensity I(x′) was calculated using Equation 63. For comparison, FIG.35B shows the detected intensity I(x′) on the CCD imaging device at theplane 742 using the original pulse of FIG. 33, which is a prior artmeasurement technique. The intensity I(x′) was calculated using Equation64.

In a third step, and as discussed above in relation to determining I(t)using the intensity correlation configuration of FIG. 24, the CCD image(FIG. 35A) is used to calculate numerically both amplitude and the phaseof the Fourier transform of the complex envelope function {tilde under(u)}_(s)(t). The complex envelope function {tilde under (u)}_(s)(t) canbe calculated using inverse Fourier transform once both the amplitudeand the phase functions of the Fourier transform are recovered. FIGS.36A and 36B illustrate the recovered symmetric temporal waveform and theoriginal temporal waveform, respectively, including the carrierfrequencies. FIG. 37 illustrates these waveforms overlaid with oneanother in an expanded time frame for comparison purposes. Thiscomparison establishes that the prediction of the pulse shape madeavailable by this invention is excellent. The discrepancy between thetwo curves of FIG. 37 is in fact a numerical artifact that can beremoved by increasing the computation accuracy.

Therefore, using the same process as described above in relation toEquation 50, the system 700 of FIG. 31 can be used to recover thecomplex envelope function {tilde under (u)}_(s)(t) of any given inputpulse using only the Fourier transform amplitude of the symmetrizedcomposite waveform. In addition, both the envelope of the intensityprofile I(t) and the underlying optical oscillations are recoverable, asillustrated by FIG. 37. Recovering both the envelope and the underlyingoscillations is an improvement with respect to the prior art intensitycorrelation embodiment described above, which only recovers the envelopeof I(t).

As described above, the waveform of an ultra-short optical pulse isdetermined in certain embodiments by using time-reversal techniques toyield a time-reversed replica of the pulse and to obtain a symmetricoptical pulse. The Fourier transform magnitude of the symmetric opticalpulse is used to uniquely recover the original time-domain optical pulsewaveform. Such time-reversal techniques utilize shorter optical pulseswith respect to the input optical pulse waveform being measured. Forexample, to determine the temporal pulse waveform of a femtosecond-scaleoptical pulse, sub-femtosecond-scale pulses are used to form thesymmetric optical pulse waveform by time reversal.

This experimental limitation of using pulses shorter than the inputpulse waveform can present a challenge which can hinder the applicationof the method in certain circumstances. However, in certain embodiments,a method determines the temporal waveform of an optical pulse using theFourier transform magnitudes of four pulse waveforms and avoiding thelimitation of short pulses. FIG. 38 is a flow diagram of a method 800 inaccordance with embodiments of the present invention. In an operationalblock 810, a sample optical pulse having a sample temporal waveform isprovided. In an operational block 820, a Fourier transform magnitude ofthe sample temporal waveform is measured. In an operational block 830, areference optical pulse having a reference temporal waveform isprovided. In an operational block 840, a Fourier transform magnitude ofthe reference temporal waveform is obtained. In an operational block850, a first composite optical pulse having a first composite temporalwaveform is formed. The first composite optical pulse comprises thesample optical pulse followed by the reference optical pulse. In anoperational block 860, a Fourier transform magnitude of the firstcomposite temporal waveform is measured. In an operational block 870, atime-reversed optical pulse having a time-reversed temporal waveform isprovided. The time-reversed temporal waveform corresponds to thereference temporal waveform after being time-reversed. In an operationalblock 880, a second composite optical pulse having a second compositetemporal waveform is formed. The second composite optical pulsecomprises the sample optical pulse followed by the time-reversed opticalpulse. In an operational block 890, a Fourier transform magnitude of thesecond composite temporal waveform is measured. In an operational block900, the sample temporal waveform is calculated using the Fouriertransform magnitude of the sample temporal waveform, the Fouriertransform magnitude of the reference temporal waveform, the Fouriertransform magnitude of the first composite temporal waveform, and theFourier transform magnitude of the second composite temporal waveform.

The sample optical pulse of certain embodiments can have an ultra-shorttemporal waveform (e.g., on the order of femtoseconds orsub-femtoseconds). The reference optical pulse of certain embodimentscan have a temporal waveform which is significantly more broad than thetemporal waveform of the sample optical pulse. For example, for afemtosecond sample temporal waveform, the width of the referencetemporal waveform can be on the order of nanoseconds. By allowing theuse of reference optical pulses with broader temporal waveforms,embodiments of the present invention allow for easier time-reversalprocesses, since the optical pulses used to time-reverse the referenceoptical pulse can be correspondingly broader. In certain suchembodiments, the sample optical pulse itself can be used to time-reversethe reference optical pulse.

In certain embodiments, the first composite optical pulse includes atime delay between the sample optical pulse and the reference opticalpulse. In certain su_(c)h embodiments, the time delay can be adjusted tomake the measurement of the Fourier transform magnitude easier. The timedelay used can be dependent upon the particular technique used tomeasure the Fourier transform magnitude. Similarly, in certainembodiments, the second composite optical pulse includes a time delaybetween the sample optical pulse and the time-reversed optical pulse,and the time delay can be adjusted to ease the measurement of theFourier transform magnitude depending on the particular measurementtechnique used.

In certain embodiments, the Fourier transform magnitudes of the sampletemporal waveform, the reference temporal waveform, the first compositetemporal waveform, and the second composite temporal waveform aremeasured using an auto-correlator as described above, or using otherholographic techniques. In certain embodiments, obtaining the Fouriertransform magnitude of the reference temporal waveform comprisesmeasuring the Fourier transform magnitude. In other embodiments, theFourier transform magnitude of the reference temporal waveform ispreviously measured and stored in memory, and obtaining the Fouriertransform magnitude comprises reading the Fourier transform magnitudefrom memory.

Using the four Fourier transform magnitudes in the same manner asdescribed above for measuring the sample nonlinearity profile, thesample temporal waveform can be measured. Note that the referencetemporal waveform can also be calculated by embodiments of the presentinvention concurrently with the calculation of the sample temporalwaveform. However, because the reference optical pulse is temporallybroad, its temporal waveform is of less interest than that of the sampleoptical pulse. In embodiments in which the same reference optical pulseis used to measure the temporal waveforms of a series of sample opticalpulses, the calculated series of reference temporal waveforms can becompared to one another, thereby providing a check of the validity ofthe measurements across the series of calculations.

In the embodiment described above, the quantities of interest (theintensity profiles of the two arbitrary ultra-short pulse waveforms) areby definition real and positive. In other more general embodiments, twototally arbitrary and different ultra-short pulse waveforms can be usedtogether to determine the pulse profiles I_(A)(t), I_(B)(t) of bothpulses. In such embodiments, the Fourier transform amplitudes of twocomposite pulse waveforms are measured. These two composite pulsewaveforms can be expressed as:I _(C1)(t)=I_(A)(t)+I _(B)(t−τ ₁); and  (65)I _(C2)(t)=I_(A)(t)+I _(B)(t−τ ₂)  (66)where τ₁ and r2 are time delays between the pulses. As described above,a classical auto-correlator can be used to generate the Fouriertransform amplitudes of the pulse waveforms. Time reversal techniquesare utilized in such embodiments due to the I_(B)(−t+τ₂) term. Asdescribed above, by using a broader reference pulse waveform, the timereversal is simpler to achieve to determine the pulse shape of anultra-short sample pulse waveform.

In certain embodiments, the temporal shape of ultra-short pulsewaveforms can be determined without using time reversal techniques,thereby providing an improved method. FIG. 39 is a flow diagram of amethod 1000 in accordance with certain embodiments of the presentinvention. In an operational block 1010, a sample pulse waveform havinga sample temporal waveform is provided. In an operational block 1020, areference pulse waveform is provided. In an operational block 1030, acomposite temporal waveform is formed. The composite temporal waveformcomprises the sample pulse waveform and the reference pulse waveformwith a relative delay between the sample pulse waveform and thereference pulse waveform. In an operational block 1040, a Fouriertransform magnitude squared of the composite temporal waveform ismeasured. In an operational block 1050, an inverse Fourier transform ofthe measured Fourier transform magnitude squared is calculated. In anoperational block 1060, the sample temporal waveform is calculated usingthe calculated inverse Fourier transform.

In certain embodiments, the reference waveform I_(Ref)(t) is symmetricand unchirped (i.e., I_(Ref)(t)=I_(Ref)(−t)). In certain embodiments,the composite temporal waveform I_(C)(t) can be expressed as:I _(c)(t)=I(t)+I _(ref)(t−τ)  (67)where I(t) is the sample temporal waveform, and χis a time delay betweenthe sample and reference pulse waveforms.

Similarly to the measurement of optical nonlinearities described above,the following relation can be derived:I _(C)(ƒ)|² =|I(ƒ)|² +|I _(Ref)(ƒ)|²+2 |I_(Ref)(ƒ)|I(ƒ)|cos(φ−φ_(Reƒ)+φ₀)  (68)where I(ƒ)=|(ƒ)|e^(jφ)is the Fourier transform of the sample pulsewaveform, I_(Ref)(ƒ)=|I_(Ref)(ƒ)|e^(jφ) ^(Ref) is the Fourier transformof the reference pulse waveform, and I_(C)(ƒ) is the Fourier transformof the composite waveform with φ₀2πfτ.

In certain such embodiments in which the square of the Fourier transformmagnitude of the composite temporal waveform is measured in theoperational block 1040, the Fourier transform magnitude squared of thecomposite waveform is expressed as Equation 68. The operational block1050 then comprises calculating an inverse Fourier transform of themeasured Fourier transform magnitude squared. For a selected time delayτ>(T_(Ref)+T), where T is the temporal width of the sample pulsewaveform and T_(Ref) is the temporal width of the reference pulsewaveform, the inverse Fourier transforms of [I(ƒ)|²+|I_(Ref)(ƒ)|²] and[2|I(ƒ)∥I_(Ref)(ƒ)|cos(φ−φ_(Reƒ)+φ₀)] do not overlap in time. Thisobservation implies that from the inverse Fourier transform of|I_(C)(ƒ)|², one can recover the inverse Fourier transforms of both[I(ƒ)|²+|I_(Ref)(ƒ)|²] and [2|I(ƒ)∥I_(Ref)(ƒ)|cos(φ−φ_(Ref)+φ₀)]distinctly.

In certain embodiments, using only the t>0 portion of the inverseFourier transform of [2|I(ƒ)∥I_(Ref)(ƒ)|cos (φ−φ_(Ref)+φ₀)], one can getthe convolution of the sample pulse waveform I(t) with the referencepulse waveform I_(Ref)(−t), i.e., Conv(t)=I(t)*I_(Ref)(−t)=∫I(β)I_(Ref)(−t+β)dβ. Taking the phase of the Fouriertransform of Conv(t) yields φ−φ_(Ref), and taking the magnitude of theFourier transform of Conv(t) yields |I_(Ref)(ƒ)∥I(ƒ)|. Since thequantities [I(ƒ)|²+|I_(Ref)(ƒ)|²] and |I_(Reƒ)(ƒ)∥I(ƒ)|are known, thefunctions |I(ƒ)| and |I_(Ref)(ƒ)| can be determined simultaneously.

Since I_(Ref)(t) is a real, even, and non-negative function, its Fouriertransform is also real and even, meaning that φ_(Ref) equals either 0 orπ. Using a Hartley transform-based algorithm as described above, φ_(Ref)can be fully determined using only the information of |I_(Ref)(ƒ)|. Notethat for a real and even function, the Hartley transform is the same asthe Fourier transform.

Once φ_(Ref) is fully determined, the phase of the Fourier transform ofConv(t) can be used to determine the phase q) of the Fourier transformof I(t), the sample pulse waveform. Thus, the quantityI(ƒ)=|I(ƒ)|e^(jφ)is recovered, and by taking the inverse Fouriertransform, the sample pulse waveform I(t) can be determined.

Usually for unchirped symmetrical optical pulses (e.g., I_(Ref)(t)), theintensity profile is gaussian-like, and the corresponding Fouriertransforms are also gaussian-like. This result implies that φ_(Ref)equals zero. Therefore, in practical cases in which the reference pulsewaveform is gaussian-like, there is no need to determine φ_(Ref) usingHartley transform based algorithms.

The method described above can be used to determine the temporal shapeof an ultra-short sample pulse waveform using a symmetric unchirpedreference pulse waveform using only one measurement (e.g., anauto-correlator measurement). The recovery of the sample pulse waveformusing the method described above is robust, even in the presence ofnoise added to the measured Fourier transform magnitudes.

In addition, once a pulsed laser system has been characterized usingthis method (i.e., the temporal pulse waveform has been determined), thelaser can then be used to characterize other pulsed laser sources, eventhough neither laser has symmetric output pulses. For example, after aninitial measurement to characterize a non-symmetric sample pulsewaveform using an unchirped symmetric pulse, one can continue tocharacterize different pulse waveforms using the non-symmetric pulsewaveform that has been previously characterized as the reference pulsewaveform for the subsequent measurements.

This invention may be embodied in other specific forms without departingfrom the essential characteristics as described herein. The embodimentsdescribed above are to be considered in all respects as illustrativeonly and not restrictive in any manner. The scope of the invention isindicated by the following claims rather than by the foregoingdescription. Any and all changes which come within the meaning and rangeof equivalency of the claims are to be considered within their scope.

1. A method of measuring a nonlinearity profile of a sample, the methodcomprising: providing a sample having a sample nonlinearity profile;placing a surface of the sample in proximity to a surface of asupplemental sample to form a composite sample having a compositenonlinearity profile; measuring a Fourier transform magnitude of thecomposite nonlinearity profile; and calculating the sample nonlinearityprofile using the Fourier transform magnitude of the compositenonlinearity profile.
 2. The method of claim 1, wherein the samplenonlinearity profile is non-symmetric.
 3. The method of claim 1, whereinthe composite nonlinearity profile is symmetric about the origin.
 4. Amethod of measuring a nonlinearity profile of a sample, the methodcomprising: providing a sample having at least one sample surface andhaving a sample nonlinearity profile along a sample line through apredetermined point on the sample surface, the sample line orientedperpendicularly to the sample surface; measuring a Fourier transformmagnitude of the sample nonlinearity profile; providing a referencematerial having at least one reference surface and having a referencenonlinearity profile along a reference line through a predeterminedpoint on the reference surface, the reference line orientedperpendicularly to the reference surface; obtaining a Fourier transformmagnitude of the reference nonlinearity profile; forming a firstcomposite sample having a first composite nonlinearity profile byplacing the sample and the reference material proximate to one anotherin a first configuration with the sample line substantially collinearwith the reference line; measuring a Fourier transform magnitude of thefirst composite nonlinearity profile; forming a second composite samplehaving a second composite nonlinearity profile which is inequivalent tothe first composite nonlinearity profile by placing the sample and thereference material proximate to one another in a second configurationwith the sample line substantially collinear with the reference line;measuring a Fourier transform magnitude of the second compositenonlinearity profile; and calculating the sample nonlinearity profileusing the Fourier transform magnitudes of the sample nonlinearityprofile, the reference nonlinearity profile, the first compositenonlinearity profile, and the second composite nonlinearity profile. 5.A method of measuring a nonlinearity profile of a sample, the methodcomprising: providing a sample having at least one sample surface andhaving a sample nonlinearity profile along a sample line through apredetermined point on the sample surface, the sample line orientedperpendicularly to the sample surface; providing a reference materialhaving at least one reference surface and having a referencenonlinearity profile along a reference line through a predeterminedpoint on the reference surface, the reference line orientedperpendicularly to the reference surface; forming a first compositesample having a first composite nonlinearity profile by placing thesample and the reference material proximate to one another in a firstconfiguration with the sample line substantially collinear with thereference line; measuring a Fourier transform magnitude of the firstcomposite nonlinearity profile; forming a second composite sample havinga second composite nonlinearity profile which is inequivalent to thefirst composite nonlinearity profile by placing the sample and thereference material proximate to one another in a second configurationwith the sample line substantially collinear with the reference line;measuring a Fourier transform magnitude of the second compositenonlinearity profile; and calculating the sample nonlinearity profileusing the Fourier transform magnitudes of the first compositenonlinearity profile and the second composite nonlinearity profile. 6.The method of claim 5, wherein: the sample has a first sample surfaceand has a second sample surface substantially parallel to the firstsample surface; the first configuration has the first sample surfaceproximate to the reference surface; and the second configuration has thesecond sample surface proximate to the reference surface.
 7. The methodof claim 5, wherein: the reference material has a first-referencesurface and has a second reference surface substantially parallel to thefirst reference surface; the first configuration has the sample surfaceproximate to the first reference surface; and the second configurationhas the sample surface proximate to the second reference surface.
 8. Themethod of claim 5, wherein the sample comprises poled silica.
 9. Themethod of claim 5, wherein the sample comprises a nonlinear organicmaterial.
 10. The method of claim 5, wherein the sample comprises anonlinear inorganic material.
 11. The method of claim 10, wherein thesample comprises an amorphous material.
 12. The method of claim 5,wherein the first configuration comprises the sample and the referencematerial in an anode-to-anode configuration in which an anodic surfaceof the sample is proximate to an anodic surface of the referencematerial.
 13. The method of claim 12, wherein the second configurationcomprises the sample and the reference material in an anode-to-cathodeconfiguration in which an anodic surface of the sample is proximate to acathodic surface of the reference material.
 14. The method of claim 12,wherein the second configuration comprises the sample and the referencematerial in an cathode-to-anode configuration in which a cathodicsurface of the sample is proximate to an anodic surface of the referencematerial.
 15. The method of claim 5, wherein the first configurationcomprises the sample and the reference material in a cathode-to-cathodeconfiguration in which a cathodic surface of the sample is proximate toa cathodic surface of the reference material.
 16. The method of claim15, wherein the second configuration comprises the sample and thereference material in an anode-to-cathode configuration in which ananodic surface of the sample is proximate to a cathodic surface of thereference material.
 17. The method of claim 15, wherein the secondconfiguration comprises the sample and the reference material in ancathode-to-anode configuration in which a cathodic surface of the sampleis proximate to an anodic surface of the reference material.
 18. Themethod of claim 5, wherein the first configuration comprises a spacermaterial between the sample and the reference material.
 19. The methodof claim 5, wherein the second configuration comprises a spacer materialbetween the sample and the reference material.
 20. The method of claim5, wherein measuring the Fourier transform magnitude of the firstcomposite nonlinearity profile comprises measuring the Maker fringeprofile of the first composite sample.
 21. The method of claim 5,wherein measuring the Fourier transform magnitude of the first compositenonlinearity profile comprises focusing a pulsed laser beam onto thefirst composite sample at an incident angle, generating asecond-harmonic signal, and measuring the generated second-harmonicsignal as a function of the incident angle.
 22. The method of claim 5,wherein measuring the Fourier transform magnitude of the secondcomposite nonlinearity profile comprises measuring the Maker fringeprofile of the second composite sample.
 23. The method of claim 5,wherein measuring the Fourier transform magnitude of the secondcomposite nonlinearity profile comprises focusing a pulsed laser beamonto the second composite sample at an incident angle, generating asecond-harmonic signal, and measuring the generated second-harmonicsignal as a function of the incident angle.
 24. The method of claim 5,further comprising calculating the reference nonlinearity profile usingthe Fourier transform magnitudes of the first composite nonlinearityprofile and the second composite nonlinearity profile.
 25. A method ofmeasuring nonlinearity profiles of a plurality of samples, the methodcomprising: (a) measuring a first nonlinearity profile of a first sampleusing the method of claim 24; (b) measuring a second nonlinearityprofile of a second sample using the method of claim 24, wherein thesame reference material is used to measure the first and secondnonlinearity profiles; and comparing the calculated referencenonlinearity profiles from (a) and (b) to provide an indication of theconsistency of the measurements of the first and second nonlinearityprofiles.